They have it both ways.
Although they are sovereign “nations,” the General Allotment Act of 1887 made Indians accepting allotments citizens of the U.S. Follow-up legislation in 1901 made all the rest of the Indians citizens as well, giving them the right to vote. via the 14th Amednment. The Indian Reorganization, Self-Determination, and Self Governance Acts have done nothing to remove the franchise from them, even though they are now sovereign nations.
Technically, their status is Dependent Domestic Entities. But, yeah, the feds see the tribes as sovereign nations. That’s why all the indian casinos are in states that don’t allow gambling.