I think Byzantium may have been pretty Roman in 477CE, but 75 years later everybody who personally remembered Rome was dead and they just had scrolls. By about the 8th century it was medieval Greek with a lot of Romanesque institutions and the "continuous Roman empire" thing was a marketing slogan.
The most Roman thing about them was that they were an empire, but you could equally well call that "Persian".
Those who looking at things from a historical timescale tend to forget that lives are short and societies change. E.g. I'm British, our empire essentially ended about fifty years ago, and anybody who seriously hearkens back to "the glory of empire" is a loony. But a thousand years from now, historical wargamers will think we were just the same as the British of 1880 and probably build forces for the reconquest of our rightful territories in America.
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