Alongside what CyborgMarx said, the Frankish Empire were German invaders rather than any sort of continuation of the Roman Empire. They also didn’t really control Rome, which stayed in the hands of the Pope.

But yes, the intent was basically for it to be the successor to the Roman Empire. The Pope crowned Charlemagne as emperor because the Byzantines had an empress regnant which the Pope saw as illegitimate and therefore saw the title of emperor as forfeited.

I don’t think he was actually called Roman emperor specifically (Nevermind, like kleeon said he was called “Emperor of the Romans”), but at that point “emperor” (well, “imperator”) had only ever referred to the Roman emperor. It wasn’t really a generic word like it is today.

Regardless, afaik it’s generally agreed by historians that the Byzantine Empire was the Roman Empire (not just a successor, but a continuation). And that the Frankish Empire (and later “Holy” “Roman” “Empire”) was not, nor was it a true successor.