I'm not here to say that transgender discrimination = discrimination women encounter based on their gender, and I don't believe that the argument that transgender people are making either. Instead, why not look at transgender people as a newly oppressed group that encounter entirely distinct issues?
Transgender people understand that they have not grown up with the same experiences as a natural born Woman, and they do not claim to be able to speak for Woman's experiences in that sense. So all they are asking for is to be recognized as legitimate people and to not be seen as suicidal, depressed, mentally ill people.
White privilege is very much a treatment by society. Racism and sexism are both enacted upon by people based on social norms.
The overarching phenomenon is ‘dominant privilege’. In a gender studies program certain people have privilege. In Japan certain people have privilege, within a ghetto certain people have privilege, within Fortune 500s certain people have privilege.
However people mainly focus on certain dominant groups in select socio economic areas of the US. But this phenomenon is found everywhere.
If a person modifies their methods of speech to better match a different identity group, for whatever reason, would they not begin to receive the level of privilege, or lack thereof, associated with the other identity group?
Now I agree that a person can't claim an "African-American identity" just by changing their skin, that seems wrong. However, if a person looks "black" then they will be treated by society like they are black, so they have to some degree shed white privilege. In the case of Rachel Dolezal, this actually worked to her benefit because of the social justice circles she was in, at least until she was unmasked. The way others treat you is related to the way others perceive your identity, not the "census definition" of your identity.
Trans women often look completely feminine. In that case, a stranger would treat them as a women, unless said stranger knew of their sex. What is privilege, if not the sum of the ways in which other individuals treat you? So, these trans women do not have male privilege, at least going forward.
Sometimes, trans women don't look feminine. They are visibly transgender. In that case, they are probably not being treated like women. However, it's glaring obvious that openly transgender people aren't treated very well by society at large. If you don't believe this, then try an experiment. Wear a wig and a dress (if you're male), or a get a haircut and "he/him" pronoun badge (if you're female), and spend some time hanging out in public spaces. Be sure to visit the bathroom. I think you'll find out pretty quickly the type of abuse that trans people face daily.
So, if openly trans people are treated badly, perhaps even worse than women, then logically they have less privilege. The radical feminist objection that they retain male privilege makes no sense.
It seems to me the ultimate goal should be to greatly minimize the differences between sexes and races (drop the traditional, ignorant differentiations we’ve created) and then the urge to identify differently to your physical expression may subside.