Indeed sex is not gender. I entirely agree.
This is why using words that have always meant sex to label this totally unconnected thing "gender" is causing so many problems.
It is true that in some parts of society, "Man" and "Woman" have reecently been partially appropriated to label personal feelings of "gender", but this is by no means at all a generally accepted meaning, and certainly does not apply to the great many pre-existing things that were labelled as "man" or "woman" using the original sex-based meanings way before the phrase "gender identity" was coined - for example single sex spaces like bogs, changing rooms and prisons, the politcal movement of Feminism, the history of female people, and women's sports, to name but a few examples.
Clearly, this new and very niche use of "man" and "woman" to label the inchoate concept of "gender" is an unhelpful and misleading, not to mention sexist, development.
In a rational world we should all, Genderists and Feminists alike, be agreeing a much better approach for all concerned is to coin new terminology entirely separate from "man" and "woman", and indeed from any originally sex-based language, to label this thing Gender that, whether or not we happn to agree is a good thing in its own right, we do all agree is different to sex.
Because as I said, appropriating other people's reality as a metaphor for ones own identity is not a "right", it is an offensive negation of other people's right to their own identities, their own self knowledge and their connection to their own history.