You need to concentrate on this fact: Like "gender identity or expression," identifying groups in the law on the basis of race, relligion and sexual orientation according exclusively to how the person identifies him/herself. So all the problems and fears and "gaming the system" that you imagine people being able to do because of the phrase "gender identity or expression," which you have well demonstrated have no basis in reality, could just as easily happen with designations of "race," "religion" and "sexual orientation" in the law.
And after all, as many people have noted, a university applicant can include or exclude him/herself from race-based admission policies ("affirmative action") simply by checking a box on a form. There is no objective method to determine if a person is "Black" or "Hispanic" or "Asian" or "white". Those are not biological or genetic categories.