The original Roman Empire did not fall till 1453, something like 5 centuries after the founding of the Holy Roman Empire.
The Eastern Roman Empire is often considered a separate entity as the Byzantine Empire, but there is no particularly logical reason to do so, as it had a continuous existence (with an intermission) from Constantine to its fall.
And they considered themselves Roman. It was a crime against
civilization that the west did not help stop the moon-worshipers from taking Constantinople.