This is a historical convention, but you are right, the Byzantines considered themselves to be Roman and referred to themselves as such.

Others referred to themselves as "Roman" too. The Seljuk Turks named their state, which they built from Roman/Byzantine territory in Anatolia the "Sultanate of Rum" which means "Sultanate of Rome".

The Latin Christians who took over Constantinople after the 4th crusade referred to their territory as the "Empire of Romania".

The Ottomans referred to their internal self-government body for Orthodox Christians as the "Rum millet" or "Roman Nation".

I've even heard a story about the post WW1 incorporation of some formerly Ottoman islands populated by Greek speaking Orthodox Christians into the Greek state. Some kids are watching the Greek troops occupy the island, a soldier asks the kids "what are you kids looking at", a kid says "We came to see the Greeks arrive", the soldier says "Aren't you Greeks?" and the kid says "No, we are Romans". (Take this last one with several grains of salt, no idea if it's true).