Gender is not a social construct. This is a meme popularized by radical feminism in the 1980s and is currently given credence by sociologists. To be honest, I don't know exactly what sociologists intend to say when they use it; but I do know what it means, regardless.
The idea is that gender is a "constructed" concept. Now, all concepts are constructed or "constructs"; so, there's no point in going out of your way to call a particular concept a "construct" except to emphasize that it isn't real. And this is the nasty part: by so emphasizing the unreality of the construct, they are in fact emphasizing the unreality of gender itself -- i.e., that gender has objective bounds.
It is easy to guess why this characterization was popular with radical feminists, who, presumably, felt confined by traditional notions of gender and thought it would free them from the prison of gender to deny the reality of gender itself. And I think this is the reason the characterization is so popular with trans people: they, too, feel imprisoned by traditional notions of gender, and to deny those notions they end up denying gender itself.
They don't realize this, of course. They only see the vision of freedom that they suppose will result from reducing gender to an artificial construct without real bounds. But, when they do this, they delegitimize themselves without realizing it -- because, if gender isn't real, then being transgender isn't real; and, if being transgender isn't real, then people who believe they are transgender are delusional.
It's natural, I think, for people who are suffering -- which certainly describes victims of gender dysphoria -- to seek any port in the storm. So, it's pointless to find fault with them for what is, essentially, a lack of critical thinking. But this error must be brought to light, because they are undermining their own legitimacy by endorsing this characterization of gender. I have spoken with some who realize this instinctively, although they can't reason it out this way. This mischaracterization is definitely problematic at a personal level, even when you don't know why.
