Native American flutes, in their present-day form, are an invention of the 1960s. The tuning was standardized on a Western-derived pentatonic scale never used before by any Native American people. So the instrument was never intended to be any one tribe's property. You are unlikely to encounter one of the pre-standardization types unless you're a museum curator.

Different peoples had different ideas about the ownership of tunes and songs. Some considered them to belong to specific tribes, others to specific individuals. There was one tribe from Colombia who expected every man to invent one tune of his own; he would never play any other one, and nobody else would ever play his.

In other parts of the world (I know about this in a Melanesian context) you also got ownership of melodies or dances by specific families/lineages; I'd be surprised if some Native American people didn't adopt that ethic.