The issue is two-fold. One, people tend to conflate sex and gender, and two, the debate hinges on nature vs. nurture, which is always polarizing. When it comes to the nature vs. nurture debate, the answer is pretty much always both. TERFS (trans-exclusionary radical feminists) will argue than any assertion that gender is biologically influenced undermines the idea that gender roles are a social construct. However, these two things don't actually have to exist in conflict.

Gender roles for the most part are learned behaviors, though we don't currently have a full understanding of the extent of this. What we do know is that there are certain biological factors that will affect gendered traits. We know that hormones, like estrogen and testosterone, are will affect certain physical and emotional characteristics, this why trans people use hormone therapy. It's also important to note that everyone experiences hormonal fluctuations, and many people will fall into different places on the masculine/feminine spectrum in their lifetime - gender is fluid. Gender is also distinct from sex (as in male/female), and people of any gender and sexual orientation may choose to present more masculine or feminine. A lot is made of the fact that trans women may present as highly feminine or adopt traditional gender signifiers, but there are also trans women who skew more masculine and eschew feminine traits (see: tech CEO Martine Rothblatt). Similarly there are cis men or women who skew feminine or masculine, respectively, and identify as their biological gender. Even biological sex is not binary, there are intersex genders. All this is to say that while gender identity is not fixed or binary, there are undeniable biological factors that can influence gender identity.

Race, on the other hand, is merely a set of superficial physical characteristics. There is no connection between the brain and racial traits. That would effectively be like saying you can find the differences between a blue-eyed brain and a brown-eyed brain. It's absurd for someone to claim that their brain doesn't match their race - the only thing that makes being black different from being white is how people treat you because of it.