Why do you think Greek-speaking inhabitants of the Eastern Mediterranean continued to call themselves Romans until the early 21st century?

It may seem peculiar to think of ethnic Greeks (as well as other groups) designating themselves as Romans, or Romaioi so long after both the Roman and Byzantine empires had disappeared, but the apparent oddity of this fact is largely a product of the conventions for naming and distinguishing certain historical groups. The inhabitants of the later eastern Roman Empire are called Byzantines by us, but in their own time they would have referred to themselves, and been referred to by others, as Romans. In our perception the Byzantine Empire eventually evolved to become an entity separate from the Roman Empire which preceded and originated it, but that evolution was not problematic, or perhaps not even apparent, for those living within the empire during any of its iterations. For them it was still the Roman empire, and their culture was still Roman as well. It may seem odd that, for example, an ethnic Greek and Orthodox Christian inhabitant of Ottoman Anatolia in the 19th century could call themselves by the same name as a Latin-speaking pagan inhabitant of Italy in the classical period. It is clear, however, that while historians might neatly delineate historical periods and people groups, those delineations must often overlook the persistent continuity of many aspects of culture; a name, a religion, or even an ethos can survive long after we might suspect.

I hope this helps you

i confess that there are several parts of this constitution which i do not at present approve, but i am not sure i shall never approve them: for having lived long, i have experienced many instances of being obliged by better information or fuller consideration, to change opinions even on important subjects, which i once thought right, but found to be otherwise. it is therefore that the older i grow, the more apt i am to doubt my own judgment, and to pay more respect to the judgment of others.