Like the Holy Roman Empire/
The "Holy Roman" Empire was all the German principalities and (for a short time) Spain and France. It was not the Roman Empire. It's name drew from the Pope Leo III proclaiming Charlemagne to be the
Imperator Augustus of Rome. But this was honorary and the Pope had no authority to name anyone emperor of anything. The Roman Catholic church did not exist when the Western Roman Empire collapsed.
Christianity existed, to be sure, but it was
Orthodox christianity, run by the Eastern Roman empire, the
Byzantine Empire, in Constantinople. The Roman Catholic church broke off from the Orthodox church after the collapse of Western Rome. It was a seperate entity frome Western Roman Empire, coming about after it's demise.
Naming Charlemagne "Roma Imperator" caused much enmity with Constantinople, because the Byzantine emperor actually carried all the power and authority of Rome, being descended directly from Constantine the Great (just not biologically). To put it in modern terms, the Byzantine emperor felt that the Pope, "was just some dude in Italy". Or worse, a heretic.
The Pope didn't care, though, and except for a short break between the dynasties of Charlemagne and Otto, kept proclaiming the protector of Catholic church as "The Emporer of Rome", until the 19th century.
Any connection to the actual Roman Empire, was in name only.