There was a question like this on the sub but I'll give an answer.

The Byzantine Empire from 476 (The end of the Western Roman Empire) till the times of Charlemagne was still called the Roman Empire as they were still Roman in Culture and in government. During the reign of Justinian the Great, his reconquests of Italy were called "Renovatio Imperii" or 'Restoration of the Empire.' The people of the Eastern Roman Empire considered the fall of Rome as a setback and that they would reconquer their lost territories.

Yet after the End of Justinian's reign, the empire seemed to become more Hellenised and slowly started to move away from being Roman. Western Europe had the same thoughts, since they didn't even have the city of Rome and don't speak the same language, yet they called themselves Roman.

Afterr Charlemagne's conquests of Western Europe, the people living there now slowly started to call him the Roman emperor since it showed resemblence to the Western Roman Empire. Then Pope Leo III in 800 crowned Charlemagne as the 'Emperor of the Romans' which was a slap in the face to state that was in name and government atleast, Roman.

Then during the Renaissance, around a century after the East fell, the term 'Byzantine Empire' was born. The Renaissance was a rebirth of Roman and Greek antiquities, so an empire which still was considered Roman would go against everything they had shown, since the Roman Empire was STILL alive. Greek peoples still called themselves Romans even after the Fall of Constantinople until the 20th Century.

I feel like this question comes up a lot but I don't seem to have saved any previous answers...except my own of course:

What did the other people's of Europe call the Byzantine Empire? Did it differ depending on who you asked and were there any changes as the empire dwindled?

Not exactly the same question maybe, but perhaps it will be helpful.