There is genuine legal, economic, and religious continuity from Rome to later Europe. When the empire fell, all of those systems didn't just evaporated but carried right in into the systems that would eventually morph into feudalism, with its many complex contractual and democratic structures. The first European ruling classes to rise after Rome were in some cases literally the same manorial powers and in others "barbarians" directly and explicitly emulating the Roman style.

Roman style class systems morphed but continued to expand, supplanting much older and more democratic forms in place throughout Europe, and it was Roman imperial law that acted as the basis to dismantle consultative and communal legal forms, meaning that Roman law became stronger and more rigid after Rome fell (in time, not immediately).

Obviously the Church remained a central power fundamentally oriented around Rome geographically and culturally, with intellectual culture almost entirely Latin until well into the bourgeois era. Rome was every ruler's source of legitimacy.

Medieval Europe by Chris Wickham is a great Marxist history of this entire process that treats the transition from empire to feudalism with a proper DHM analysis.