I think gender identity in its current form is a social construct. That is 'gender' has become the defining aspect of the real person rather than the objective world. That the sense of self has some spiritual or transcedent realness in the world.
I think the development of 'gender identity' has been happening for decades. I think it began with critical feminism which expanded into LGB studies and the around the 90s with Queer theory.
M<ost of the pioneers in queer theory were prominent lesbian feminists. I remember Judith Butlers quote that gender is performative which means that there can be no gender identity before the gendered acts, because the acts are continuously constituting the identity.
I think this is one aspect of gender identity that came out of feminism that gender identity was in large part performative. So it seems at least in part gender identity is an expression of how one feels about themselves as performed within society. One can if inclined choose to act feminine or masculline.
What degree of performance is socially constructed or natural is something that would need to be destinguished. I don't think they are completely socially constructed.
For example studies have shown that infant male chimps will tend to play with toy trucks and blocks similar to humans. There are some differences in thinking such as males relating to spatial aspects and geometric shapes which females relate more to social aspects, facial expression and emotions.
The question is could a society or an individual flip the traditional traits of men and women and say recondition a male to act feminine without some underlying inclination is hard to know. But I do thing we can perform without being inclined that way.
But I think at the same time society has cultivated the idea identity as being the measure of success. I think is has been accellerated by social media and reality TV. Everything is about how someone presents, their lifestyle, their looks and cloths and cars and how many likes.
This I think is another idea from the feminist movement that the "political is the personal". Politics moved into the personal domain and it wasn't about good policies but how they presented and identified with the people. Marketing a created identity that people would relate to. Thats why we could see politicians being remodeled according to the electorate.
Social media has given a voice to all now and so everyone is an identity with something to say and look like. Identities are then divided by race, gender, sex, lifestyle, looks, wealth, success, how happy, healthy, who you know, or what crazy feat, ect. Everyone is then grouped as the bad and good identities. Such as white males being the lowest on the identity hierarchy.