I still don't think 'Gender' exists. I think personalities exist (1) , and that one could refer to them as 'genders' (lower case) or 'genres' or 'types' if one didn't want to use 'personalities', and that of those possible terms' 'gender' is the worst precisely because it is a grammatical term associated with language elements called 'masculine' and 'feminine' so people start to imagine it is more linked to sex than to anything else, or even, as now for so many people, that it a replacement for sex.
Our sex is part of our personality. And the number of different personality types is indeed infinite. The old days when we looked up our star signs or our Myers Briggs personality types (so many of them, trite though they are) seem innocent now compared with the gender juggernaut.
Our society may have favoured stereotypes around the sexes, though they are going to be very variable through time and place. People marketing products will certainly try to convince us that our personality / genre / type / star sign / nationality / gender / nationality / class / whatever means that we will be particularly happy and admired and fulfilled if we buy or wear or eat or use whatever it is that they're selling, and we really ought to look and behave like the actor or model they've chosen to represent people like us.
And all that is probably worthy of a bit of sociological analysis if you like that sort of thing, and people, particularly young people, who get very hooked on the advertisers' dream models probably need some psychiatric advice. Better keep the psychologists away from them, they're probably either under the sway or in the pay of the advertisers.
But I don't think 'Gender', that essentially sex-based set of predictive stereotypes, is even as real as star signs.
(1) The horrible thing about saying personalities exist is I fear I may be agreeing with Jordan Peterson, but it can't be helped.