There seems to be a spectrum of views about man/woman. At one end, you have the biological determinists, who argue that a man has a penis and a woman a vagina, and that's that. This is often used to attack trans people.
At the other end, there is the whole argument about social construction, which used to be said about gender but is now also said about sex. The most extreme here, I suppose, argue that 'woman' and 'man' are fictions, also the idea that gender is a masquerade.
Somewhere in the middle there is a kind of 'social self-representation' idea, that men and women present themselves in certain ways, e.g. clothes, hair, make-up, and so on.
There is also the issue of personal history, i.e. experiences that we have gone through as men and women.
As NS said a while ago, we don't have a language really to describe trans people, but we don't really have a language to describe sex/gender. It's striking that 'gender' itself is now used to describe sex identity, which maybe shows how it is all changing fast, and categories melting.