It is a very American thing (nation of mostly immigrants) but I don't see how claiming to have Native American heritage is functionally any different from claiming to have Greek, German, or anything else heritage. Especially if it's true. I get that this is being played out for political reasons but this was supposedly part of her family's beliefs and part of their family story.
Maybe it's different for us as we have a somewhat similar family heritage that is all over the place. Maybe the difference is that I can point to an ancestor on the Dawes rolls and my DNA test has a much higher percentage of Native than does Warrens but we also don't claim tribal citizenship or affiliation, only heritage.