Based on my reading of The Inheritance of Rome by Chris Wickham, the short answer is that in the early medieval period, there wasn't a particularly strong conception in the lands of the Western Empire between a 'Roman' and 'post-Roman' era. Obviously this was even less of the case in the Eastern Empire, but I assume OP is mainly asking about the West.

In part, this is because the Western Empire fizzled out quite slowly and in an amorphous fashion, and wasn't replaced by what we might call 'coherent' nation states. Mediterranean trade - the lifeblood of the Empire - also declined terminally but very slowly. There was no set date for the Fall of the Empire as we conceive now - the Sack of Rome in 410 would most likely have been received with the same sense of shock that 9/11 was felt throughout the Western world. But at no point was it conceived as marking the end of the Roman world as 'Rome' at this point was centered in the East, and within the Italian peninsula the city of Rome had long been in decline.

Above all, 'Europeans' would also have conceived of themselves as members of Christendom, from which we can draw a pretty straight line from Constantine. So its for this reason that Charlemagne conceived of his Empire as a continuation of Rome, and not some sort of revival.

Related question: How widespread was knowledge about the Roman empire during that time? The lack of formal education and literacy would probably mean most people would only hear the odd tale passed by word of mouth?