I remember as citizenship of poleis was something that foreigners could be awarded...
Even among Athenai people differentiated themselves between demos, for example with trophies related to military campaigns in other countries...
SPQR had so many social/legal divisions, I don't think the whole "empire" considered themselves as Romani, even Lucan during the principality said he was a Celtiberos...

In ancient times, family had a much more important role than today, I can easily see commoners referring to themselves with patronymics or tribe names...
Only distinctions I think were social classes and whether or not someone was a member of a royal family...