With 90% of the American people claiming Cherokee ancestry--why have Native Americans has such a hard time getting basic civil rights?

It has to do with 96% of statistics on the internet being made up out of hand.

Na, there has been a long standing tradition of Americans claiming a little Indian blood. There was never any shame in that. White people claiming Black blood has never been as prevalent. That ancestry has always been downplayed, even though the levels are just about the same, or even more common in many regions. You'll almost never hear a White family claiming to have Black blood and they certainly don't theorize the possibility or desperately search for this connection. You'll never hear a Whiter person claim a GG grandmother that was "said to be part-Black" and then want to assert this as their "African heritage" in the same fashion that they do this with bogus Cherokee blood family lore. There was a racial hierarchy established in this country with Africans on the bottom. So, a lot of these family myths are covering for Black-White admixture, and many are just made-up out of thin air. People like the idea of Cherokee blood.

But, most of these myths are exaggerated or incorrect. It's not in the 90% range, but the percentages of these claims is very high nevertheless. In the Black community, Indian blood was claimed to downplay White admixture.

This also doesn't mean they support tribal sovereignty and self-determination. Usually, this just means that they are more aggressive about attacking Cherokee national status. They'll be upset that their bogus CHerokee blood is not accepted and they also use it as rhetoric. Such as: "I have some Cherokee blood, and I don't need a card to prove it." Or, "My family is part-Cherokee and we are just American. Why do you have to be separate now. We are all just Americans." So, they'll diminish the need for the nation or community to recognized Cherokee families and to determine Cherokee tribal citizenship, and they'll usurp this "heritage" based on nothing but family lore that is not proven (as if that is all there is to being Cherokee, just claiming "blood").

UPDATE: This is a non-facetious answer.