The Greeks called themselves Romans as it was synonymous with Christians, but they acknowledged their pagan past. See people such as Anna Comnena. Westerners routinely called it 'empire of the Greeks', as did the Vikings. That other guy is intent on downplaying the Greek character of the Byzantines and is obsessed with referring to them as classical Italian Romans. I don't know his agenda.
According to wikipedia:
"Byzantine Empire" is a term created after the end of the realm; its citizens continued to refer to their empire simply as the Roman Empire (Greek: Βασιλεία Ῥωμαίων, tr. Basileía Rhōmaíōn; Latin: Imperium Romanum),[2] or Romania (Greek: Ῥωμανία, tr. Rhōmanía), and to themselves as Romans. "
"The Byzantine Empire, also referred to as the Eastern Roman Empire, or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople (modern Istanbul, formerly Byzantium). It survived the fragmentation and fall of the Western Roman Empire in the 5th century AD and continued to exist for an additional thousand years until it fell to the Ottoman Empire in 1453 "
https://en.wikipedia.org/wiki/Byzantine_Empire
Not claiming that they are classical Italian Romans, but it was a continuation of the Roman empire but run out of Greece. It didn't have anything to do with Christianity and a connection the the papacy in Rome (whereas the Holy Roman Empire did). The Byzantine empire was Orthodox, with their own locally administered church.