if gender is a social construct anyone can identify as that gender.
This is such a profound mistake that so many feminists make. It must be a trick of language that makes so many women (including some 'gc' feminists) believe something so demonstrably false.
Gender means 'sex roles'. It is socially constructed upon the material reality of sex. It is not separable from sex. Superficial markers of femininity are not the same thing as sex roles, and men do not experience sexism when they adorn themselves with said markers.
This is obvious, glaringly so. It is proven, in many, many contexts all over the world. From the Hijra of Pakistan who have much more freedom and autonomy than Pakistani women and girls, to Lily Madigan getting away with behaviour that would be tolerated in no Labour woman, to underachieving Travis Alabanza being featured alongside black women of actual achievement in Evaristo's 'womxn' issue of the Sunday Times. None of these males are 'treated like women', because no male can be; none of them have ever been subordinated by sexism, though they may of course have suffered as males. The gender 'woman' cannot be transferred to men via lipstick, or hormones, or pronouns, or surgery.
Again, this is obvious, this is proven, this is demonstrable. Why do so many women continue to believe the opposite, in contradiction of the evidence? Is it wishful thinking? Effeminate men are not 'like women', they are not treated like women, they have no special insight into the female experience, and their behaviour in general indicates that they don't empathise with women, at least, no more than other men do.
I am a woman, female, and not remotely feminine.
But I bet you have experienced sexism nonetheless. I bet you are still 'treated like a woman'. Sex-role non-conforming women often suffer the worst sexism (witness the horrific violence inflicted on butch lesbians in parts of the world, and their near-total invisibility in popular culture). The gender 'man' cannot be transferred to women who defy sex roles.
'Gender' (weasel word) is a phenomenon that is inextricable from sex; it is not an independent entity that can be shifted between the sexes according to how femininely or masculinely someone presents. As long as a person's sex is discernible to others, he or she will be treated as their actual sex.