I agree with you there. I would not have said anything if the original lady had not identified her father as "native american"(which by bloodline would make her talking about her genetic heritage) i felt it was inappropriate for the second lady to correct her, but of course this second lady was probably only listening for trigger words anyway, since that seems to be the theme of the times. And i realize that everyone has trigger words. I have them as well. Mostly my experience is that native americans usually identify themselves by tribe, if the topic does even come up beyond a general statement.
I may be mistaken, but my opinion is that these two addresses are a northern/southern differentiation.

My husband is Native and that is exactly what he calls himself, “Native American.”