Your question is definitely interesting but also exceedingly complicated, as I gather there is no real consensus on what constitute "roman identity", especially concerning the ethnicity of it.

So it brings up several questions as to "what constitute a roman?" , "what did they themselves regard as 'roman' when they called themselves such" and so on.

There's been several theories that try to explain or bring light unto this issue - some more popular than others - but there's certainly many interesting reasons as to what would define a roman citizen as exactly that, roman.

What most agree on is that "roman" as in something more than a citizen of the city of Rome was mostly something that began to develop in Augustan times and after that. Before the advent of the Roman Empire, the republic gained territory much more piecemeal and through alliances -- the empire brought greater, much extensive landgrabs and fostered the idea of orbis Romanus - the roman sphere or world, the idea that Romans where to conquer and it was their right and prerogative for it. This establishment of the roman oecumene, its territory, is widely thought to have been the reason for a shift towards Rome outside merely its city and allies. The oecumene became as much as a word for the idea of "where civilization (Roman empire) was" as much as to say where you would find Rome.

One theory proposed in order to create a "roman identity" was through the method of creating a sense of solidarity. The example argued here is that roman citizenship was now given outside the boundaries of its city, something that was very rare in antiquity. This could be argued to have created a form of loyalty towards the emperor by having him grant regions citizenship rather than just in Rome, the city. Problem of this would be to ignore the association between the emperor and capital, which could be argued to be the same thing

Another theory is that of humanitas. The roman concept of this word doesn't quite mean the humanities of the Enlightenment which we now associate it with. But it is the amalgamation of traditions, human virtues and qualities that corresponds to a set of "ideal type" -- this ideal humanitas would be used to judge the 'barbarians' and 'un-romans' by comparing them up against this type, but just as well to compare how a roman would be tasked to govern and rule over them. In a way, one could argue that the roman humanitas served as a symbol to roman dominion over others and how they were to be formed around it.

Some would argue against just that however, pointing out that "romanizing" was not an active policy of Rome nor truly given much thought to it. An argument is that many of the people that was under roman territory never truly made more roman, but it was a unintended consequence of implementing roman military and political hierarchy system into conquered territories which through these complex and ultimately roman systems one would gain an identity by it.

If we however go straight to a primary source we can observe that Tacitus' work Germania might point out a key factor for creating a sense of roman identity outside Rome, which would be urbanism: the ideal of a settlement system organised around a laid out urban center which fulfilled a range of public functions, such as political organisation, religion and state administration. Towns with the fora, baths, temples, public buildings. Even the people living outside of the colonia, city, were dependent and partook in the system of the city and therefore came to share the ideology of urbanism.

Many of these theories overlap and some might not even be mutually exclusive. One could very well argue the idea of a roman identity outside that of being a citizen of the city of Rome was a gradually changing concept. There are arguments for saying that romans even during the republican era was never to considered an ethnically composed group in the modern sense. They would point to the concept of shared myths which the romans had quite a few of; for instance Virgil's story of Aeneid and the idea Rome connected with the Trojan fables, the rape of the Sabine women and the asylum of Romalus on Capitoline Hill which served as an temple of refugees from the outside all might argue that Rome was always a city of immigrants and more welcoming of customs and tied its identity across any ethnic lines.

To conclude, I'd say that tendency of Imperium Romanum to be less about the imperium of the city of Rome but that of romans across the imperium is something of a gradual shift from Augustan times and outwards. The book I am sourcing explains that there is a definite lack of studies and work done in regards to how one would identify as a "roman" outside Rome and that it is hoped that this will be rectified in the coming years.

Sources

Roman identity in Byzantium: A critical approach

Ways of Being Roman : Discourses of Identity in the Roman West

From the Byzantine sense, they may argue that Rome, the state, never did cease to be tied to Rome, the city. An important component of the Byzantine political thought is that of translatio imperii.

Take the Patria, a 6th century work by Hesychios Illoustrios, which begins with the following proclamation:

“When 362 years had passed since the sole reign of Caesar Augustus in the elder Rome, and her fortunes were already coming to an end, Constantine son of Constantius took over the sceptres and established the new Rome, ordering that it should be equal in rank to the first.”

The passage serves to illustrate quite evocatively how the foundation of Constantinople, New Rome, was seen and interpreted already in the early Byzantine period.

An ailing "elder Rome" that was rapidly declining in relevance was vicariously 'reborn,' in a sense, through the foundation of the new Rome as instituted by Constantine. This passage illustrates that the claim of translatio imperii served as an important component of Byzantine political thought against the charge of a "Rome-less Roman Empire."

In the Byzantine ideological perspective, they did indeed hold "Rome," as Constantinople itself was the New Rome. The 'elder' Rome in Italy did hold continued interest by the Byzantines in its historic significance, but through translatio imperii, Nova Roma had become just as 'Rome' as the Rome of old. Anna Komnene illustrates that forwarding this argument additionally served geopolitical purposes, as it meant all the authority of old Rome, including ecclesiastical, were thus transferred by Constantine to Constantinople.

Sources:

• Alexiad 1.13.4.

• Berger, Albrecht. trans. 2012. Accounts of Medieval Constantinople: The Patria. Cambridge.

• Matheou, Nicholas S. M. 2016. “City and Sovereignty in East Roman Thought, c.1000–1200: Ioannes Zonaras’ Historical Vision of the Roman State” in eds. Nicholas S.M. Matheou, Theofili Kampianaki and Lorenzo M. Bondioli, From Constantinople to the Frontier: The City and the Cities. Leiden.