This has already happened to male people, it's not new. Joseph Boyden, the writer, comes to mind.
It's rampant in academia in North America. In part because academic posts are difficult to come by and often it's an advantage - officially - to have a minority designation even if it's not related to the position - it's totally legal in Canada to do this and it happens in the US as well. It's not just people faking, but also people with some indigenous (or black) ancestry that they don't really have a connection to, who are polishing it off for job applications - that kind of thing goes on quite openly among young academics looking for positions. (And not just there, I've known families who needed to prove French ancestry for access to French schools and to call the connection tenuous is generous. But everyone accepts that this is the way you get access to special services and rights.)
It's likely to ramp up IMO. As more and more significant advantages and rights are being attached to indigenous identity in particular- taxes, jobs, fishing quota, etc - indigenous groups themselves are more and more wanting to have clear definitions of who is eligible. I suspect in many cases it's going to end up as something closer to a citizenship model (which already exists for some groups) rather than DNA as such.
I suspect the shit will hit the fan around the whole thing once we get to that point and hope I am not around for it.