James K's answer is excellent, but I want to expand further on how tribes are organized presently.
However in the 21st century this appears to be quite an archaic concept, since in theory all citizens are supposed to be equal rather than receiving certain privileges simply because of their race or ethnicity.
I know you are from Europe, so you may be unfamiliar with the history of reservations/reserves in the US and Canada, but this is a misconception of the arrangement. ("Reserves" is the Canadian term, but I'll use "reservations" because I'm American.)
It's also a little unclear which privileges you refer to. The only one I can think of associated with reservations is the ability to run their own court system, which is of course, available to every municipality, county, and state in the US, so far from limited to tribal governments. In fact, their court systems are more limited than others, as they cannot prosecute non-members.
The point is, though, that the arrangement exists not between the federal government and US citizens of certain descent, but between the US federal and state governments and the "domestic dependent nations" that represent Native American/First Nations peoples. This concept is known as tribal sovereignty.
Tribal sovereignty in the United States is the concept of the inherent authority of indigenous tribes to govern themselves within the borders of the United States.
America is sometimes considered to be a nation organized around shared ideals, rather than ethnic history, like most other countries are. (There was a good Politics question about that, but I can't find it.) However, tribal nations are, an issue that causes no end of contention within the nations (look up
Even though the US is set up this way, it can still have relationships with ethnic nations, and it does.
So it's best not to consider the relationship not one between America/Canada and its citizens of Native American descent, but one between the respective governments and their dependent (ethnic) nations.
Simply put, disorganizing a reservation would be disorganizing a nation with (limited) sovereignty, and perpetrated by the nation(s) that stole their land and forced them into the reservation in the first place.
* All that said, this is very heterogeneous. British Columbia has a lot of unceded land and unsigned treaties. Alaska has a different set-up than the lower 48, which I'm not qualified to discuss, but the notion holds at a high level for why the reservations continue to exist.