Once upon a time, there were a bunch of city states whose populations had some shared ancestry, close enough dialects that they could understand each other, and some shared religious ideas and mythology. They didn't consider themselves the same people as each other, but they still considered themselves something closer to each other than other societies around, so they were the Greeks, and the others barbarians.
But then one of them conquered the rest, and his son Alexander conquered a lot of other people, and a more unified identity started to form. They started thinking of themselves as a common people with a common culture, and maybe they'd have a shared destiny.
But then a different group took them. Armies from Italy calling themselves Romans, a group that might have once been considered a multi-ethnic people, but the other Latins and even the Oscan and Etruscan were already fading into this identity. And the Greeks tended to get along well in this empire. Caracalla made them all citizens in the early third century AD (along with all other freeborn men in the empire). A century after that, authors tell us that if you asked most Greek speaking people in the empire what they were, they'd say they were Roman.
Things got a little weird when the Western half of the empire fell. Greek was already dominant in the eastern part of the empire, but it started making more sense to shift laws from Latin to Greek, since far more Romans could speak that language when the Westerners weren't Roman anymore. And for a while, the West called those Greek speakers Romans as well. But Western their stories about the history of the Romans became more legendary and idealized, and as the remaining Roman empire shifted in culture as all people do, they started to doubt their "Romanness." The pope ended up crowning some guy named Charlemagne as the new Roman Emperor, much to the confusion of the people who had been the Romans for centuries. And since it didn't make sense for Westerners to be the successors to the Romans when there were still Romans, they reverted to an old name for those guys over out East: the Greeks.
The Romans who spoke Greek didn't consider themselves the same as the Romans who spoke Latin's descendants (and later German), and they started talking about those Latins as uncouth barbarous people who weren't like the Romans at all. Still, they called on those primitive Latins to help them reconquer some territory from the Turks, and those Latins took it much farther than intended and kept much of the former Roman lands they took. Things fell apart between the Romans who spoke Greek and the Westerners who considered themselves the real successors of Rome, culminating in the Latins conquering the Roman capital Constantinople in the Fourth Crusade. Those conquerors declared themselves the Roman Empire. The surviving Greek speakers who set up government elsewhere insisted they were still the Roman Empire. And the Germans called themselves the Holy Roman Empire. None of them contained the city of Rome.
Those Greek speaking Romans whose distant, distant ancestors considered themselves Greek did retake their city, but the empire slowly declined until it was taken over by the Ottomans. But those conquered people still considered themselves the Romans. And there were those who wanted independence for the surviving Roman people. But that needed Western support, which was difficult as the Westerners didn't like those "Byzantines" and thought that culture was decadent and the people slimy and treacherous. But do you know what Westerners liked? Those extinct ancient people called Greeks.
And so those working for independence started insisting they were the people of Pericles, Leonidas, and the glory of an ancient, idealized past. And plenty of those now being labelled Greeks hated it. They were Romans. Proud orthodox Christian people following the glory of Constantine, Justinian, Heraclius, and Basil II Porphyrogenitus. They had nothing to do with those ancient pagans. They didn't even speak the same language! You could get trilingual dictionaries that translated between Greek, Latin, and "Romanish," the language that descended from Greek that modern Romans spoke.
But being Greek brought support. And propaganda started changing minds. The people whose parents called themselves Romans started liking those old Greeks. They stopped talking about them the way we talk about the Ancient Egyptians. Now they were their ancestors and the originators of their culture. Romanish was just modern Greek. And that big empire that lasted after the West fell? Totally Greek. Not Roman at all. After all, if those people were Roman, they wouldn't be the ancestors of the people who were now obviously Greek.
Things like nations and ethnicity aren't just constructs, they're weird and far from static.
