How can feminism and transgenderism both be true?

If gender is not determined by physical characteristics, then what is it, exactly?

Someone recently told me that transgenderism is “in line with current scientific understanding“. Maybe so. But I’ve also been told that it’s “in line with current scientific understanding” to believe that masculine and feminine ways of acting and dressing, and “gender roles” such as mothering and bringing home the bacon, are mere cultural constructs.

Also that it’s sexist to believe that men are better than women at some things that don’t depend on physical size and strength, such as math and science. In short, the radical feminist dogma is that the only differences between men and women are physical differences. Otherwise, we’re equal in every respect.

If all the non-physical characteristics that we commonly associate with the respective sexes are mere constructs or the product of chauvinist attitudes, then what’s left? What is there, besides physical traits, with which a person may constitute a valid and objective “gender identity”?

But if we admit that subjective gender identities are real and legitimate, then don’t we have to admit that there are real differences between the sexes apart from physical differences?