Well, pretty much everyone liked to claim himself the True Heir of Rome. How much such varying pretensions now really mattered (outside official propagandas of course) is another question.
Yet, the Byzantine claim is by far the best of them.

The state itself was very much Roman, but it evolved over the years, eventually ditching Latin as the official language in favour of Greek, wich is what the vast majority of the population had as first (and often only) language anyway.