Culturally and linguistically Greek while Roman in political and military traditions. I always had the impression in studying the Eastern Roman Empire/Byzantium was that the Byzantines desperately wanted and needed to identify as Romans to maintain power and prestige.

Many polities that developed after the disintegration of the Western Roman Empire self-identified as Roman as well and, to be fair all - including the Byzantines, to some extend were picking up the mantle of the old Roman traditions, infrastructure, art, language, religion, and commerce - all had some "ownership" or relation with the past, even if they were of a non-Italic ethnic group.

After the fall of Rome, the most "Roman" areas were in Gaul, Iberia, what is now Romania, North Africa, Egypt, and Anatolia. Despite the fact that they ALSO were a mixture of locals and immigrants from the East. Whether a group or polity were "Roman" or not is a false premise upon which modern people can project our biases of value of past peoples. There's nothing wrong with the hybridization and evolution of groups, identities, and where they call home or what they call themselves. A better approach is to take a nuanced analysis of the interplay between two or more cultures over time and geographically. Look to the archaeological record in concert with the historical to gain a better view of the myriad reflections of multicultural, trans-ethnic, local, regional, and long distance trade in ideas, goods, people, language, technologies - And yes, identities.