I've always wondered about this case seeing as many Greek people I've met tend to associate the Byzantine Empire as a part of Greek history. This is understandable as Greece was an essential part of the Byzantine Empire at its zenith. However, I wondered where those who lived in the Byzantine Empire back in the day identified as Greek or not? Their roots are from the split between the Eastern and Western Roman Empire, and, if I recall correctly, the Roman Empire aggressively conquered Greece into their empire.
Belonging to Byzantium was never really a matter of blood. It was a ‘commonwealth’, a racially mixed group of what we would today call Slavs, Bulgarians, Serbs, Greeks, Thracians, etc. Greek wasn’t the only language spoken though it was by far the dominant one.
But they certainly didn’t see themselves as Greek- at least not as we use the word today. ‘Hellene’ had a pagan connotation- they called themselves Romanoi- which meant ‘Roman’. The term "Byzantines" or "Byzantine Greeks" is an exonym applied by later historians.
Greek-speaking Christians continued to refer to themselves as Romans even during Ottoman rule. It fell out of fashion in the late nineteenth century and the rise of the Greek nationalist movement.
I've just realized the question should say "themselves" instead of "itself." In any case, thanks for any future answers!