The concept of translatio imperii — the transfer of imperial authority from one polity to another — was the dominant framework for legitimate rule in medieval Europe.
The Holy Roman Empire claimed to be the legitimate successor to Rome. Charlemagne's coronation in 800 CE was framed as a translatio. The Tsars of Russia called Moscow the "Third Rome."
What's interesting is that the framework required the previous empire to have ended. Continuity would have meant subordination. The claim was always: we are what they were, but better and renewed.