I would say that people have used gender terms in some form or other across time and culture. Without necessarily being aware of it in that way.
Quite some people across the political spectrum from the one extreme up to the other occasionally or at some point have used a gender term that they believe exists.
Some of these are, woman, man, beta male, alpha male, "not a real woman", "not a real man" and so on.
And some people even extremes that would call themselves alpha male. Probably are alpha male in terms of gender expression. But gender fluid (going from beta to alpha) internally.
For sex you can choose to define it as
woman (sex) = person with xx chromosomes
man (sex) = person with xy chromosomes
Definitions can't be true but they can be useful. So for different uses we can have different definitions. Surely for medical and scientific and epistemic reasons the old definition is useful.
For social reasons a different is also useful. However if we use the same word. We are begging for equivocation fallacies and for the opposite of what language is meant to do. Namely to help us navigate and understand each other clearly. (any person less likely to exert thoughtful cognitive effort show the effects of this and I don't want to be condescending either since it is understandable)
Feminine is now often woman (gender).
Now the chromosome based definitions is strong it for example explain why a woman (sex) is still a woman (sex) if she loses her breasts. And why a man (sex) is still a man (sex) if he becomes a eunuch. A eunuch would be a man (sex) without a penis.
Additionally
Gender was used way before we used it.
When a lesbian or "masculine" or just neutral woman was told to act more feminine. Or to act as a woman (even though she obviously was one). Or when the Godfather told Johnny Fontane to "act like a man". Even though Johnny obviously was a man.
Clearly The Godfather didn't mean that Johnny was not a man (sex) but they did mean that Johnny wasn't acting in line with the cultural/personal idea of how a man (sex) should act. But he does say it in a descriptive way. So he wasn't acting like a man (gender).
So this is important. It's nothing new really. We just became more aware of it. But there's a potential drawback in defining gender without ant reference point.
Ofcourse if one were to say that a man (gender) is someone that wears pinks clothes with lace thong and has a vagina and was born with a vagina. Then that wouldn't really be in line with the common idea of what a man is. Neither sex not gender.
We can accept someone saying this for moral reasons. But for other moral reasons we must recognize that if this is how we generally used language we'd be in the dark. And so most people don't use language in that way.
Some frame of reference is helpful. Surely we can dismiss of genitals in the gender concept and focus gender on secondary or some primary traits. And we can cherry pick them as we please (meant in a good and respectful way). But the less we have some kind of essence the more likely people will be confused.
I mean a person born with xy chromosomes and a penis (man sex). Who feels they are a woman (gender). And wants to express as such. In general will not want to express and transition to become a person that looks as what most people would call an alpha male. So some essence (be it without the need for genitals) is helpful and very reasonable.
So to conclude. Gender can be filled as we want but some essence is healthy.
Some would try to mingle sex and gender and I have noticed that this happens on either side of the debate. (Which we luckily can still have).
I think it's useful to recognize that at some point we have to draw a line somewhere such that we can point to things in reality with words such that we understand eachother.
Even if hormones are ultimately what makes us man or woman (sex). Since that is what DNA ultimately is meant to bring forth. And even if it is difficult to detect exactly where the distinction is. Especially then it is helpful to use genetics as a way to resolve it.
Surely if a virus breaks out that requires a different vaccine for people with xx chromosomes and XY chromosomes we would want to be safe enough to use such words. And equally we should be safe enough to have words for how one feels or wants to express.
However I do think it is possible to be wrong about oneself. If I were to say that I have two legs. Then the only thing that makes that objectively true is me having two legs. If I have one leg then the proposition is false.
If I a man (sex) were to say that I am a woman (sex) that's not possible. Given the definitions with chromosomes.
If I a man (sex) were to say that I am a woman (gender). Then it's reasonable (even a duty by clinics?) to perform a Socratic questioning method.
If I were to say that for me woman (gender) is someone that wants to wear female clothes. Someone that wants to be with men. Wants to wear makeup. Wants to have breasts and so on.
Then whether I am a woman (gender) depends on whether I am the person that wants those things which I just gave as my hypothetical personal definition that aligns with the words in the proposition.
But if I were a man (sex) and I were to say that I am a woman (gender). And when asked what is that for you? I respond with. A woman (gender) is a person with xx chromosomes, with a vagina.
Then if we are equally epistemically virtuous we can recognize that if my statement is to be true. Then it would require me to have xx chromosomes, and a vagina. So in that hypothetical the statement would be false.