Ethnicity as a topic can be quite difficult because of modern interpretations of the word versus how the Romans actually interpreted the concept, but I will try and highlight some evidence for you. The Romans, especially during the Republic, did see themselves as of a different from those within Italy. Before and after Rome conquered or absorbed the 'ethnic' areas into their sphere of influence, they would have saw themselves as different - especially given that there was significant differences in terms of culture and things like language (Etruscan itself is thought to have been very different to Latin despite the small geographical distance). It is a difficult topic begin the focus is on the city rather than the "nation" - consider the alliance of the italian cities as a collection of communities that understood strength in numbers. During the Second Punic War Italian communities sided with Hannibal in the south, which shows the cultural differences that existed.

Even after the Social War such ethnic distinction played a role within Roman society. A quote from Cicero (Letters to Friends. 15.20.1):

My orator-for that is the title I have given it—I have handed to your Sabine servant. His nationality made me think that he was a proper person to whom to give it: unless he too has availed himself of the licence of candidates and has suddenly adopted this surname. However, the modesty of his look and the gravity of his conversation seemed to me to smack somewhat of Cures. But enough about Sabinus.

Here you have Cicero suggesting that political candidates had adopted a Sabine name in order to give them a political advantage because of the typical qualities associated with the Sabine people.

Within Latium, members of a family that had its origin as being from around Rome (ager Romanus or Latium vetus) had an advantage over those from places that were conquered within the region (Latium adiectum). Cicero (For Plancio. 19) writes that Iuventius had an advantage over his client Plancius because Iuventius was from Tusculum and Placius from Atine, the former being older and closer to Rome:

You are of that most ancient municipal town of Tusculum, from which many of our consular families are derived, among which is also the Juventian family; there have not so many families of that rank proceeded from all the other municipal towns put together. Plancius comes from the prefecture of Atina; certainly a less ancient and distinguished abode, and not so near to the city. How much difference do you think this ought to make in standing for an office?

Such ethnic roots are witnessed even on coins, Farney in his book provides information on coins. He identifies 43 coins with ethnic advertising, with 26 making reference to towns within Latium vetus, and then 13 make reference to a Sabine origin.

Hopefully this answers your question by showing that there was ethnic distinctions at the time that were understood. The topic of identity and ethnicity is one that has produced a lot of scholarly work. I would recommend these works if you had interest in further reading.

Farney, G.D. (2007), Ethnic Identity and Aristocratic Competition in Republican Rome, Cambridge.

Hölkeskamp, K.J. (2010), Reconstructing the Roman Republic: An Ancient Political Culture and Modern Research, Princeton.