Not true. Most of the benefits of "sovereignty" apply only to members of the tribe (ie, members of that ethnicity) who are operating on the location in question.
I'd point out, though, that as time goes on it gets harder to state with any kind of certainty that someone who claims to be an "Indian" isn't telling the truth.
Simply explaining how the ethnic group identified as Iroquois Indians is organized should give you a complete understanding.
First of all, there are Confederation and Non-Confederation Iroquois. The Non-Confederation people are usually known as Mingo, and their language, which is indistinguishable from Iroquois, is called Mingo. There are several websites where you can learn it. You will frequently hear it spoken on "The Simpsons" (for reasons I do not understand).
Within the Confederation, way before the American Revolution there were 5 nations. At some point in time families within each constituent tribe were selected to travel together to the Carolinas to colonize that area. They were called Tuscarora. Later on when the Tuscarora returned to New York they were given various pieces of land in existing tribal territories.
During the American Revolution two tribes elected to assist the United States in the war against the Brits. I believe there were a couple of neutrals, and the other tribes elected to assist the UK.
At the end of the war all the tribes, winners and losers, were protected by the same Treaty of Paris that recognized American independence.
The Congress wrote a special law recognizing and protecting its allies among the Iroquois.
later on reservations were set up ~ but that's way after the Revolution.
In the meantime New York discovered they could get away with simply stealing Iroquois land ~ first they declared that the Oneida were white people with no right to live on the Oneida lands, then kicked them all out, and took the land which was sold, essentially, to illegal aliens.
Later on New York took on the tribes protected under the Treaty of Paris and began forcing land cessions with little or no payment.
Finally, they faced the Seneca and Mohawk, special protectees of the English Crown, who had reservations on and across the Canadian border, and they weren't able to kick them out or steal their land. That's where the cigarette tax stuff was invented. New York seeks the same thing ~ the destruction of the Iroquois ~ only the trick changes over the centuries. This is still a trick, and the objective is the same.