I think it was meant very much in the figurative way. In continental Europe, Roman society never really stopped, it just changed a lot. Even Goths and Lombards would try to "become Romans" as much as replace them, happily adopting titles and structures (with varying levels of success).

A big hindrance to seeing continuity is that popular knowledge of "romanness" is so focused on the late republican/early imperial era and there is surely not much visibly continuity between early medieval and a snapshot of the Roman state outdated by several hundred years.