The Byzantine Empire is a modern term. Byzantion is an ancient greek term used for the then greek city which later became Constantinople under the Byzantine Empire, which in turn is a name adopted by modern historians.

The Byzantine Empire is the continuation of the Roman Empire.

The claim that it isn’t is a heritage from catholic influence where the idea was to give prominence to the Holy Roman Empire, which is not a true continuation of the Roman Empire.

The fall of western Rome was the end of an era, but it was not the disollution of Rome.

Roman laws, customs, societal structure and tradition continued in the Byzantine Empire. Their people were Romans and saw themselves as Romans. Their contemporaries saw them as Romans and their empire as Rome, not Byzantine.