Oh, there are quite a few studies about gender and they've even found that the brain structure of transwomen is closer to that of ciswomen than cismen and vice versa. But in my opinion that only means what we call gender is just a collection of characteristics that come from a certain brain structure.
But these studies have all been conducted within a society with a gender concept and therefore can't prove that gender is a thing that exists without society. In a society without a concept of gender - or at least without the concept of gender roles being somehow tied to biological sex - people would just be accepted with their unique collection of characteristics and nobody would feel like they were born into the wrong role and thus into the wrong body.
Basically, without gender there wouldn't be any dysphoria and no transgender people. If nobody tells you that the combination of characteristics that makes you you is usually found in people with a p**is you wouldn't even get the idea that you "should" have one yourself.
Once again, the human need to put everything into neat categories came back to haunt us. First, everybody was put into two convenient boxes. Then people noticed they were in the wrong box. Further study reveals there aren't even enough boxes, creating more and more of them. And slowly some people are starting to realize there aren't any distinct boxes at all but rather just a giant continuum of everything society for some reason felt the need to draw little boxes onto. I'm using present tense in the last few sentences because we're still in that era of creating new boxes and figuring out that reality doesn't think in boxes at all.
