Just a cursory look over history and you see how many cultural things are traded and etc. It seems untenable for folks to claim a thing, let alone figure out who then gets to claim it considering the vaugnesses of race and backgrounds.

I also worry that the effort to clam ownership or whatever we want to call it actually just creates more isolation.

A local educational institution used a local native american symbol for a long time. It was always done respectfully (no weird chanting or anything). One tribe even regularly sent drummers and dancers to open sporting events. It was a really cool thing to see / be a part of, everyone had a great time in a community that didn't have much in the way of connections between the tribal entities and etc.

But it was decided that the local tribes should vote on if it should continue or not. One tribe overwhelmingly voted yes. Another tribe due to tribal politics chose not to vote. So out went the tradition, away went what was one of the few connections between the reservations and non reservations. I think it was a great loss.