Why are we led to believe that we can choose our gender but not our race when both are dictated to us by genetics?
I think the premise in your question is slightly wrong, although I also think people are generally really confused on these topics.
But for example, many people who support transgender rights activism would not agree that transgender people are
choosing their gender. They believe that the mismatch which transgender people feel between their assigned gender and their self perception is natural, something that is just intrinsic to their sense of identity. They aren't choosing to be transgender any more than you are choosing to be cisgender.
On race, anthropologists and social scientists would disagree with the statement that race is dictated by genetics. For example, the anthropologist John Relethford writes that race is a "culturally constructed label that crudely and imprecisely describes real variation" (
2009). This topic gets a little complicated in terms of how racial categories "crudely and imprecisely" describe real variation, but the point is that the racial categories we use are not based in accurate measures of genetic difference. The size of the genetic variation between two people identified as "black" is often much larger than between either individual and others who we identify as "white", because there is more genetic variation between African sub-populations than between other sub-populations. If racial categories were determined by genetics instead of skin color, they would be different. The other obvious evidence for the lack of a solid genetic basis for race is simply that racial categories change over time.
But, what I think you are accurately getting at in your post is the idea that race is an
ascribed status. We don't get to choose it for ourselves. Other people look at us and decide what race we belong to. The categories are dependent on cultural background, but they exist objectively, they are independent of our self-perception. That's also obviously true of gender as well, and if that were not the case then the existence of transgender people would not create any social issue. So another mistake I think people make is confusing "social constructionism" as individual choice.
I think the difference between gender and race, at least in relation to transgender people, is that gender categories are much closer approximations of biological realities (sexual dimorphism) than racial categories are. There really is no such thing as "feeling white" in a biological sense. The idea of "feeling like a white person" only makes sense against some cultural background, because the biological basis of race is especially weak. The biological basis of gender (in sexual dimorphism) is much stronger, but we also know that there are lots of interesting ways in which biology can vary and that people can have a combination of sexually male and sexually female traits, both at birth and in later development (see
this thread for some links). It makes more sense, then, for people who have been ascribed male sex to legitimately feel like they are female, as the result of some biological considerations.