We should be suspicious of subjective things like what women say.
If a person identified as left-wing, would they be left-wing?
Quite possibly, because political beliefs are abstract.
If a British person identified as French, would they be French?
No, because nationality goes beyond individual belief. But there are things they could do to become French, e.g. apply for citizenship. Abstract identities can be changed.
If a human identified as a cat, would they be a cat?
No, they would be a person who identified as a cat. They can never actually be a cat because of biological reality.
Do you see the difference?