Ridiculous. Gender is not "constructed"
Of course not. But this idea has been pushed in academia for decades now. This is the way societal changes occur. First the ideologies become accepted in academia; then within decades they diffuse throughout society, eventually becoming common wisdom.
You're basically right, but the technical details tell a revealing story. Gender is indeed constructed by man. But people (and animals) don't have gender, since gender is just a characteristic of words. La porte—French for "door"—is a noun of feminine gender. Human practice, rather than God or nature, makes this so.
That's why homo-activists and assorted other malcontents absurdly try to apply the word "gender" to flesh-and-blood creatures—to imply that sex is a mutable decision made by human beings (or perhaps goats), rather than a trait built into every cell of a creature's body.
The malcontents pretend to believe in "gender," as opposed to sex—even though no one does. The reality is too obvious. But my observation is that the people who go in for the fiction are so angry at the world for their own fears and shortcomings that they'll make claims for human "gender" to create a safe zone of unreality for themselves. So the idea is more than false. Like other forms of Marxism, it's more of a devout lie, with all the power hunger and hatred for one's neighbor that that implies.
There's a reason this sort of b.s. is taken seriously only in places like universities and welfare states such as New York City, which are so subsidized as to foster their own little zones of unreality.