People who lived in Rome (or the people who worshipped the Roman gods) called theirselves Romans. Even tough know the religion is dead is there still a Roman ethnicity? I am asking this because even tough the old Greek religion is now dead there is still a greek ethnicity but I don't see anyone calling themselves romans.
Follow-up question : as a roman myself (ie born in Rome from a mostly roman family), I would be very interested in some DNA research on the lineage of the roman inhabitants through the ages. For instance, how much did the great invasions of Italy during the 4th and 5th centuries altered the "original" genome (already a mix of greek- italic and etruscan people) ?
Roman-ness was a legal status rather than an ethnic one. You should check out these old threads on Roman citizenship:
Roman citizenship and voting rights
Did The Roman Empire Have Universal Citizenship in the Late Antiquity?
How was citizenship verified in ancient Rome?
TL;DR: In the very early days of the Republic, Roman citizens were basically the free-born inhabitants of the city itself. As Rome began to conquer first Italy and later the rest of the Mediterranean, they began to extend the rights and privileges associated with Roman citizenship to certain allies they wanted to reward: local elites in conquered territories, or soldiers who served in the army for 20 years or more. Being a citizen enabled you to petition the government, hold certain public offices, be tried under Roman law, and required you to pay taxes. Citizenship was eventually extended to every free-born inhabitant of the empire --- but considering the amount of slaves in Roman society, that certainly didn't encompass every person inhabiting Roman territory. You could born in Gaul or Thrace, never set foot in Italy, not speak a word of Latin, worship only your local gods, and be Roman. Equally, you could be born on the Palatine Hill and not be Roman, if you were a slave.
Google search actually brings up a relevant askhistorians thread by /u/gorat.
Was there a Roman ethnicity in the first place? It was fairly cosmopolitan, both as a city and certainly as an empire.