I think it may be more prudent to reword your question along the lines of “how did Roman subjects in the various parts of the empire identify themselves, throughout the history of the Republic and Empire?”

You’re coming with an understandable assumption that the Empire would have instilled a sense of national identity for all under Roman rule, as most modern states operate, but this was far from the truth for much of Roman history.

While more can be said of the entirety of Roman history and stretching into the Middle Ages and the Byzantines, or to the far flung reaches of the empire from Spain, Britain, and Constantinople, I can speak a little about the time during the Republic and slightly beyond.

During much of the Republic, the Romans, citizens, were those who lived in the city of Rome itself. Those who lived outside of Rome on the Italian peninsula, subjects, were considered Latins, without the rights and privileges afforded to citizens of Rome. You have to remember that the story of the Romans is one of conquest- the Romans in antiquity didn’t unify the societies surrounding them to all become Romans (the Etruscans, Samites, etc.), they conquered them, and either extinguished them or loosely ruled them, extracting taxes.

It is also important to remember how Rome conquered peoples, typically allowing them to continue with their cultures and religions, so long as they paid taxes. When would a conquered people begin to think of themselves in the identity of their conquered, if at all, especially if their conquered did not integrate them or afford them citizenship? To complicate things even more, Rome would eventually also often send out settlers to newly conquered lands. How do these settlers identities incorporate into these societies, when do people on the Iberian peninsula begin to think of themselves as true Romans? When do the Iberian settlers start to feel less Roman and more something else? Do the Romans in the mainland think of people in these far reaching parts of the empire as Romans or something else?

The Roman identity is one that had changed over many centuries as the Romans conquered more lands and different peoples and began incorporating aspects from other cultures into their society (ie. Importing Greek slaves for tutors), and vice-versa, who constituted a Roman citizen would change, and the importance of Rome itself, the seat of power, would change.

On who constituted a Roman citizen during the Republic we can look to the Gracchi brothers for some insight. The Gracchi brothers have an exceptionally interesting history of norm-busting that very arguably eventually led to the rise of Sulla, Marius, Caesar, and Augustus and the demise of the Republic- but that’s a story for another time. One of the main campaign arms of both Gracchi brothers was the status of citizenship for the Latins. They wanted citizenship afforded to the Latins outside of Rome, among various other reforms such as tax and land reform, and went as far as bringing Latins into the city to support their political causes which caused quite the stir. This proposal, granting Roman citizenship and rights to non-Romans was, to say the least, a hot-button issue. The Gracchi brothers, in their separate times, used the mobs to shout down the Senate and achieve their ways, for a time being. The Senate and Roman ruling classes vehemently rejected the idea of affording citizenship to these outsiders. Eventually both Gracchi brothers were murdered and their bodies thrown in the River Tiber, and also eventually the Latins were afforded citizenship.

All of this is to say though, it’s not as simple as building a national identity, because that wasn’t the Romans prerogative. While we may believe it beneficial to build a national identity today, that we are all -insert identity here-, such was not necessarily the case for the Romans. Part of the power of being a Roman came from the exclusivity of the identity itself.

And indeed as Rome expanded and successive emperors spent less and less time in the city of Rome itself, starting i believe with Hadrian, what it meant to be Roman, and the cultural and political significance of the city of Rome itself became to wane and be questioned.

The declaration of Constantinople being the capital of the Roman Empire would have been unthinkable to the Romans of the Republic because to them, only Rome was Rome, and only the Romans of the city were Romans.

So, it’s not as much a question of when did people stop thinking of themselves as Romans, but what did it mean to be Roman as time crept on? How did this identity evolve, how did conquered subjects over time think of themselves, how did the settlers maintain their culture, how did the citizens in the city think of those in the provinces, when the emperor never set foot in Rome was he still a Roman? When the seat of power begins to shift around from Ravenna and Constantinople, is this still Rome?

The messy, gradual, disintegration of the Roman Empire, along with its culture, and how far flung and de-centralized the empire was make these all very difficult questions to answer, especially as time passes and the empire gets messier and messier. (I haven’t even touched on the importance of religion in Roman society and how we consider Rome and Romans as it begins to Christianize as the Roman faith was a foremost hallmark of the Roman identity.)

Hopefully that provides some insight on how this is by no means a straightforward question with any sort of straightforward answer, but that there was this constant push-pull, evolving dynamic of what it meant to be Roman and what was Rome in the first place.

And hopefully, someone can come along with more expertise than me and shed some better light on the issue ad I’d love to read that as well.

On the last question - some Pontic and Cappadocian Greek still considered themselves "Rhomaioi" well into the 20th century (and yet some considered themselves different from Hellenes). As explained eloquently in other comments, they probably meant something different than people who claimed to be Romani 2500 years earlier on the Apennine peninsula.