I'm no expert on either topic. But I believe humans basically start off as female in the womb, and either become male or don't. And there are many intersex conditions. The body responds to hormones typically associated with either sex. So gender is fluid in a biological sense. If someone transitions to male, female or nonbinary, they already kind of contained that potential.
However, race is a social construct, usually based on heritage as well as biological appearance. So it's hard to say how much biology is really involved. Does the human body contain the ability to be any race? Or to cultivate an appearance that prompts other humans to socially categorize you as one race or the other?
Maybe for people who are mixed race, there is a sort of spectrum available to them. They likely know how to present themselves in a way that gets them categorized as one race or the other.
But otherwise, not really. If you're White, and you say, "I identify as Black," the question might be: do you have Black heritage? If you don't, you can't really create it out of thin air. There wasn't a situation while you were in the womb where various hormones could have influenced you to appear more Black than you do. If your parents are both White, they were going to have a White baby, no matter what. Race is a social construct, but it's based on appearance and heritage. It's about belonging to a group, not about being an individual, the way gender is.
If you're assigned female at birth, and you say, "I identify as male," then cool! Your body already has the capability to become hormonally male. You can socially identify as male. Any human, of any race, has this potential. Any two parents could have a baby that is any sex or gender, depending on various factors.