Broadly speaking you cannot draw an equivalence between Roman Empire and Americas because the dominant ethnicities in US, Canada, Brazil, Mexico and other countries are from outside. But IMO this would be the closest analogy.
The equivalent would be if the Italian Greeks had become the dominant ethnicity in Italy and absorbed the Romans, Samnites, Etruscans, etc into their culture.
The native Americans would absolutely consider themselves as more similar to each other than to the European and African arrivals. Just by matter of undeniable physical features.
As for the Romans, it is within the mists of history and mythology (Rape of the Sabine Women), but nobody can deny that there was an original people there in Italy, and other peoples who became citizens and/or subjects of Rome weren't ethnically Roman.
Once you loose the knowledge of your ancestry that connection is lost forever, until the end of time.