There may have been a reason for that: back in the days when there were miscegenation laws, they varied from state to state, but in many states, someone who was “black” (1/8 or more) couldn’t marry a white person, but if your dark skin was due to being native American, even fully so, you could marry a white person in those same states. If you were 1/4 black, and had straightish hair and a long nose, and wanted to marry a white person, putting “Little Bird,” or something as your middle name on the marriage license application, so the clerk wouldn’t even ask, made perfect sense. Which is to say, it was through-the-looking-glass behavior, when that was where you were. You can hardly blame people for doing what they needed to do when faced with things like miscegenation laws.
That said, I get what the OP is talking about. Every now and then, we get someone at the synagogue who had one Jewish grandfather (or great-grandfather), wasn’t raised with anything Jewish in the home, and may even have been raised in another religion, who demands to be treated like a regular Jew with no need to convert, who ought to be eligible to be synagogue president tomorrow, ought to be called to Torah, even though he knows no Hebrew, and it goes on.
It would make me mad if I grew up on a reservation, and lived an authentic life, which meant that I didn’t have some privileges, and someone came out of the woodwork, and demanded to get the same few benefits I might also have from being authentic, and he had some thin trace of ancestry.
It’s one thing to say that you have an native American ancestor, and another to assert that you are Cherokee, or whatever, which I would understand to mean tribal membership. I mean, I have a single Irish great-great-great-grandparent (maybe just two greats-- I can’t remember), so I can say I have some Irish ancestry, but I’d be lying to say I’m Irish, or even that I’m Irish-American.