> Does this mean those are random "1/16th native american" WASPs and such, that claim the heritage because of the results, but otherwise didn't know about it, or have any real relation (demographic, cultural, etc) with native americans?

In the theory under which “race” is meaningful, it is biological and distinct from (and orthogonal to) social “ethnic” categories, so claiming race on that basis is exactly as valid as the idea of race itself.

Now, if we admit that “race” is just a dishonest label assigned to certain ethnic categories...