Came here to say much the same thing. A great book that gets into the details of this is https://www.amazon.com/Inheritance-Rome-Illuminating-Dark-40...

TLDR; people considered themselves "Roman", or something adjacent to it, long after the western empire officially disintegrated, and of course in the east it didn't really die.

Even on the outskirts (e.g. Romano-British) people held onto Roman, Christian, and Latin culture at a very deep level. And even when the Anglo-Saxons invaded they were extremely aware of who and what the Roman ruins etc. were, even if they chose not to settle in them in many cases.