but suppose i decide that i am a black male, internally. it seems to me that race is a physical characteristic like sex, and therefore follows the same legal rules, or should.
You only have half the puzzle here.
Race and sex are both appear, on the surface, to be collections of physical characteristics. However, race is actually a social construct and the characteristics that are used to separate people into races vary from culture to culture and even historically within one culture.
Meanwhile, even without considering gender identity there is a difference between the various possible sex genotypes and phenotypic characteristics. In short, sex is a strange conglomeration of the actual genes, primary sex characteristics, secondary sex characteristics, gender identity, and societal gender roles.
The big difference between the two is that the human brain has an innate sense of maleness or femaleness that is more than just determined by social constructs. A person with male genitalia and a female brain will see herself as a female because she has a female brain. Whereas race is completely constructed: there is no known innate sense of whiteness, blackness, Japaneseness, etc. The delimiters on race are thus products of those same social constructs and cultural differences.
A person using a bathroom corresponding with their gender identity is doing so because their brain is literally programmed as "male" or "female," and the person is using the bathroom that appropriately expresses their masculinity or femininity. Society using its constructs to deny that identification puts an unnecessary psychological burden on that person. Whereas things such as scholarships for black students are based off the consequences of historically harmful social constructs. Even if you identify as "black," you have not been affected by the same historical racism or institutional discrimination that black people have, and thus you should not be the beneficiary of responses to those cultural issues.
Likewise, citizenship is completely a social construct that has no physical characteristics at all associated with it. There is no such thing as being born with a Russian brain because Russia is not a thing that actually exists in nature; it's something that humans made up. So no, believing you are Russian has no impact on whether you are currently a US citizen, because the citizenship rules are completely based on external forces and not at all on your own biology.
As for thinking you're a tiger, there is no current evidence of a species identity in the brain, nor is there even a plausible neurophysiological explanation for how an alternate species identification would develop. As such, with current knowledge, those identifying as another species are mentally humans who have some sort of psychological
reason for thinking they're animals, which puts it in a completely different ballpark. Indeed, if your brain literally developed as a tiger's brain, you would not be able to tell us "I am a tiger" because tigers can't talk and type. Legally, you should still be recognized as a human and thus not barred from entering cities, and still held culpable if you decided to, say, maul your coworkers. Whereas if you truly acted like a tiger you would probably be declared incompetent to stand trial (since you'd be effectively mentally disabled compared to the standard human) and likely would become a ward of the state since no one could affirm that you are, in fact, a tiger.