They don't "make me a woman" but those social interactions are a pretty fundamental element of the experience of being a woman.
Would you make that same statement to a black person, who repeatedly comes back to the challenges and barriers they face as a black person being an intrinsic element of their lived experience and their identity as a black person in the US?
Yes I would, absolutely. Carlton was as black as Will on the Fresh Prince of Bel Air.
"I'm not really an X unless I'm **** on for being an X" is a hooooooorrriiibble mentality. You can link your self identity to your downtroddeness and ever expect anything to get better.