The paper "Sex Assigned at Birth" by aTRA attorney published in the Columbia Law Review in 2022 has plenty of information on this topic, and it's heavily footnoted so you can check the original sources yourself to make sure the paper is accurate.

The paper makes what in my view are a lot of wild, crackpot claims - such as asserting that scientists do not agree that the concept and term "biological sex" have validity and that the phrase "biological sex" was originaly invented and came into use to promote transphobic, homophobic and heteronormative biases, beliefs and practices.

But all that said, the paper does attempt to trace and show when and how the terms "sex assigned at birth" and "assigned sex at birth" and the thinking behind them came to be adopted by TRAs and believers in gender identity ideology generally.

I've pulled out passages that I think deal directly with your question, and I've woven some material found in the footnotes into the text:

Transgender rights discussions often turn on the distinction between “gender identity” and “sex assigned at birth.” Gender identity is a person’s own internal sense of whether they are a man, a woman, or nonbinary. “Sex assigned at birth” means the male or female designation that doctors ascribe to infants based on genitalia and is marked on their birth records. Sex assigned at birth is intended to displace the concept of “biological sex.”

This Article provides an account of the origins of the terms “biological sex” and “sex assigned at birth” and assesses the potential of the shift to sex assigned at birth for transgender rights arguments.

The debate is not one over mere nomenclature. This Article’s examination reveals that the term “biological sex” rose to prominence to lend a veneer of scientific support to projects denying the validity of transgender identities and that the unquestioned use of that concept continues to underwrite exclusion.

By referring instead to sex assigned at birth, transgender rights advocates convey that “biological sex” is not simple, static, or binary and that gender identity also has biological aspects. Furthermore, the phrase “assigned at birth” invokes philosophical arguments against assigning particular social roles to individuals at birth. It taps into the moral intuition that a person’s genitalia and health data are private matters.

This Article argues that sex assigned at birth is an important concept that clarifies the stakes of disputes over transgender rights. But it cautions that this conceptual shift is not sufficient to secure victories in transgender rights litigation.

In the last few years, the concept of “sex assigned at birth” has appeared with increasing frequency in U.S. case law on discrimination against transgender people.1 The phrase had been used, at least since the 1960s, to describe an obstetrician’s “casual pronouncement of the newborn as a male or female,” “based upon inspection of the external genitalia.”2 This pronouncement, then and now, results in a male or female designation on a child’s birth certificate that is sometimes considered the person’s legal sex, unless changed through formal processes.3 Over the past two decades, the concept of sex assigned at birth has been taken up by transgender rights advocates to replace the troublesome term “biological sex.”4

Assigned sex... is the basis for the legal definition of “transgender”: having a gender identity that does not match the one expected for an individual’s sex assigned at birth.6 It appeared in 2016 regulations interpreting the Affordable Care Act 7

Footnote 7: Nondiscrimination in Health Programs and Activities, 81 Fed. Reg. 31,376, 31,467 (May 18, 2016) (to be codified at 45 C.F.R. pt. 92) (defining “gender identity” as “an individual’s internal sense of gender, which may be male, female, neither, or a combination of male and female, and which may be different from an individual’s sex assigned at birth”).

The idea of assigned sex popped up in discussions of transgender individuals in the 1960s, when American physicians adopted the term “sex re assignment” to describe surgical procedures for transgender patients.75

The term “reassignment” appears to have been borrowed from discussions of surgical procedures for people with intersex variations.76

The use of assigning language in this context may have been part of a larger debate in the medical community at midcentury about whether transgender identity was a type of intersex variation resulting from biological causes that should be treated with surgery or a psychological malady resulting from early childhood development that should be treated with psychotherapy.77

This shift to assigning language may have entailed an implicit critique of the idea that original birth certificate designations are authoritative pronouncements of “legal sex.”78 One contributor to a 1969 collection of essays on medical treatment of “transsexuality” referred to the practice of assigning sexes on birth certificates as “arbitrary” and “superficial.”79

Doctors advocating for sex reassignment surgery in the 1960s distinguished “sex assignment at birth” from other definitions of sex, such as psychological sex, social sex, sex of rearing, anatomical sex, and legal sex.80

Some medical professionals, however, resisted a multifaceted understanding of sex in favor of the fiction that sex is a simple matter of X or Y chromosomes—a fiction that seemed to serve the purposes of legal rules that distinguished between men and women. For example, in New York City in the 1960s, a few transgender women who had socially transi- tioned and undergone hormone therapy and gender-affirming surgeries were able to persuade city bureaucrats to change their birth certificate sex designations.82 In 1965, the New York City Board of Health convened a committee of medical experts that decided not to allow any further changes on the ground that “male-to-female transsexuals are still chromo- somally males while ostensibly females.”83

In 1970, the idea that sex “assignment” was legally relevant was rejected in Corbett v. Corbett, an influential English case on how marriage law pertains to transgender individuals.91 The question in Corbett was whether April Ashley, a transgender woman, counted as a woman for purposes of her marriage to a man [named Arthur Corbett, an Eton-educated British aristocrat in line to inherit the estate, peerage and title that would make him the 3rd Baron Rowallan]. 92 The court concluded that the validity of the marriage depended on Ashley’s “true sex,” a question on which it heard conflicting testimony from several medical experts.9.

Ashley did not dispute that she was born [male], but argued that because her psychological sex was female, she “should be classified medically as a case of intersex and that since the law knew only two sexes, male and female, she must be ‘assigned’ to one or the other which, in her case, must be the female.” 94 However, the court was not persuaded that “transsexualism” was a type of intersex condition, due to the testimony of experts that “transsexualism” was “a psychological disorder after birth...

The Corbett court concluded [in the April Ashley case] that “the biological sexual constitution of an individual is fixed at birth (at the latest), and cannot be changed, either by the natural development of organs of the opposite sex, or by medical or surgical means.”98. The court regarded Ashley’s vagina to be “artificial,”100 rendering sex with her husband indistinguishable from homosexual activity.101 It rejected Ashley’s argument that her female “assignment” should count for purposes of marriage because, in the court’s view, Ashley had “confuse[d] sex with gender.”102

Although [the court ruling in the case of April Ashley's marriage to] Corbett did not use the term “biological sex,” that concept began to trickle into U.S. case law in the 1970s, used almost exclusively in cases involving transgender rights.104

In the 1980s, the term “biological sex” appeared in [US] army regulations on the exclusion of gay men, lesbians, and bisexual people from [US] military service.105 Defining sex as fixed by biology at birth was a way to reinforce homophobic policy because it prevented, for example, one member of a same-sex couple from using sex reclassification to evade the ban on same-sex marriage.106

In the 1980s,[US] courts continued to use terms like “biological sex” almost exclusively in the context of debates over LGBTQ rights.107 An infamous Seventh Circuit opinion in 1984 held that transgender individuals were not covered by the sex discrimination provision of Title VII because “sex” means only “biological male or biological female,” an attribute that cannot be changed.108

By contrast, beginning in the 1980s, the medical community increasingly used the term “assigned sex” to replace anatomical sex in discussions of transgender patients.

The American Psychiatric Association’s influential Diagnostic and Statistical Manual of Mental Disorders (DSM) be-gan to classify conditions related to transgender status as disorders in 1980.110 In 1987, a revised version of the third edition of the DSM defined gender identity disorders as “an incongruence between assigned sex (i.e., the sex that is recorded on the birth certificate) and gender identity.”111 The term “assigned sex” replaced the term “anatomic sex” that had appeared in that passage in the 1980 edition.112 This change seems to have been intended to broaden the definition to apply regardless of whether the individual felt that their identity was inconsistent with their anatomy.113

However, the change was not made consistently: Although the DSM’s diagnostic classifications in 1994 and 2000 used assigned terminology,114 its glossaries referred to “biological sex” rather than assigned sex in defining transsexualism and gender dysphoria.11

In the 1990s, transgender theorists reappropriated the concept of sex assignment in an effort to define their community...

Virginia Prince, who is sometimes credited with coining the term transgender in the 1970s, defined it [the new term transgender] as “somebody who lives full time in the gender opposite to their anatomy,” as opposed to a “transsexual,” meaning a person who has sought medical intervention to alter their anatomy.117

In the 1990s, scholars and activists struggled to find an “umbrella” term to describe the coalition that included transsexual, transgender, and many gender- nonconforming individuals, among others.118 These scholars and activists began using “assigned” terminology, rather than anatomy or biology, to define various identities under the transgender umbrella.119

Nonetheless, the phrases “sex assigned at birth” and “assigned sex” did not appear in a handbook by leading experts on transgender rights legislation published in the year 2000.123

The term appeared in [some] law review literature on transgender rights prior to that time, in the context of discussions of intersex variations and as one in a list of various medical definitions of sex. See Julie A. Greenberg, Defining Male and Female: Intersexuality and the Collision Between Law and Biology, 41 Ariz. L. Rev. 265, 269 (1999) (listing several definitions of sex developed by doctors in the context of intersex variations, including “assigned sex” in an argument for transgender rights).

[But in the 1990s] those terms appeared in only a small number of reported judicial opinions.124

[In fact] courts in the 1990s began referring to “biological sex” with increasing regularity.125 The first U.S. Supreme Court opinion on transgender rights, Farmer v. Brennan, decided in 1994, referred to “anatomical” and “biological sex” rather than assigned sex, and labeled a transgender woman who had not completed genital surgery as “biologically male.”126

in the early 2000s, transgender rights advocates and legal scholars advanced a shift toward the language of assigned sex rather than references to anatomy, biology, or other aspects of the body.129

In 2002, Philadelphia and New York City amended city laws to forbid discrimination based on gender identity, regardless of “sex assigned at birth.”130 The state of California adopted similar language in 2004.131

By 2006, the editors of a collection of essays on transgender rights began by explaining that the term transgender “is now generally used to refer to individuals whose gender identity or expression does not conform to the social expectations for their assigned sex at birth.”132

Although the 2013 DSM V used the term “natal sex,”133 that term appears infrequently in case law [in the USA during the same time period] 134

In most cases before 2015, [US] courts [that used "assigned at birth" language] did not seem to regard the distinction between assigned sex and anatomical sex to be of any import... Most of them simply quoted some other source using the phrase... In a few of these early cases, courts regarded assigned sex to be different than the anatomical sex of a transgender person who had undergone surgery, with the transgender person’s current anatomy providing the better indication of their current sex.

[In one case from Minnesota in 2007, the state court concluded that a [plaintiff who had undergone sex reassignment surgery and changed birth records had changed her sex and “[i]t would be wholly inappropriate for this Court to invent a narrow federal definition of ‘sex’ based on the sex assigned at birth and impose that construction on a Minnesota statute”); Creed, 2007 WL 2265630, at *1 (referring to the DSM’s term “assigned sex” but still noting that “[i]t is unclear whether the plaintiff is still biologically a male”).

[But] After 2014, a year often described as a “tipping point” for public acceptance of the transgender rights movement,references to assigned sex began to proliferate in the case law.136

This shift was likely driven by the efforts of advocates for transgender plaintiffs to use the term ["sex assigned at birth"] in their briefs and distinguish it from “biological sex.”137 For example, a 2015 complaint filed by the ACLU on behalf of Gavin Grimm, a transgender boy who was denied access to the boys’ restrooms at school, used assigned sex terminology138 and noted its disagreement with the school district’s definition of “biological sex” by placing the term in scare quotes.139

During this same period, the competing term “biological sex” began to pop up in more legal and policy contexts. The term “biological sex” had appeared infrequently in biomedical research prior to the 2010s, but its usage increased over the course of that decade.140 In 2016, it appeared in a much-discussed North Carolina law excluding transgender people from restrooms consistent with their gender identities.141 Around that time, similar legislation was proposed in many other states and localities, offering various definitions of “biological sex” based on criteria such as reproductive organs or genetics.142 These varying legal definitions reflect the fact that there is no agreed upon definition of “biological male” or “biological female” in the sciences.143

https://columbialawreview.org/wp-content/uploads/2022/11/Clarke-Sex_Assigned_At_Birth.pdf