- In the East side of the empire, Constantine, the Roman emperor who moved the capital to Byzantium ("New Rome") was half-Greek, and the Greek element in the East meant this half of the Roman Empire had a stronger Greek ethnic presence.
- In the West side, the local Roman elite along with newly arrived Germanic peoples (the Franks) were Christianized and established the Papal states, of whose the Pope was king, the Catholic church, and realms that continued from the Roman Empire that was split into West and East. To make their claim over the Roman Empire stronger, there were fabrications of legitimacy (See https://en.wikipedia.org/wiki/Donation_of_Constantine ) and a lot of religious infighting with the East.
In short, in terms of a continuum of emperors the Eastern empire was essentially uninterrupted.
Even before the breakup of the empire, Romans were a bit obsessed with lineage and being descendants of powerful Romans. This cultural element carried over in the next two millennia by many people claiming the role of the Emperor of Romans, until the victories of Napoleon forced the rest of Europe to dissolve the Holy Roman Empire in 1806 to prevent Napoleon from claiming the title for himself. https://en.wikipedia.org/wiki/Dissolution_of_the_Holy_Roman_...
