Its not a factual identity because thats not what culture is. Culture is the lie we tell our children so they will not stray from the group. Just as Hitler and the Nazis created the Aryan culture, Chakotay's people would create an identity that allows them to isolate themselves from the multicultural blend of the federation and the culture of starfleet.
Imagine if that 'native american revivalist' faction leads the push to isolate earth from the Federation. It would be as bad as the earth first group from the enterprise series.
And then Chokotay discovering his native american symbology on an alien world in the Delta quadrant reveals that that native culture has alien influences anyway.
Chakotay (having discovered it was all a lie): "Nooooo!"
If you were looking for a darkness within Voyager and the Star Trek Universe it is there.