In their own minds, they never stopped being Roman. Their emperors traced their lineage back to Augustus. Many of the trappings of power and the court were Roman in origin. If you asked one of their soldiers which army he served in, he would have said the Roman army, possibly wondering if you were an idiot. Remember that the very term "Byzantine Empire" was a much later, Western, construction and would have been unknown to the citizens of that time.

Some important things did change. The language of the territories comprising the Eastern Roman Empire had been mostly Greek, and eventually the Imperial court gave up using Latin (there was little point in its use other than tradition at that point). But for these changes, it might be better to say that instead of the ERE becoming less "Romelike", that "Romeness" was now defined by the culture coming out of Constantinople.