I don't know much about Italian History post Roman Empire, so Micchan will have to give you that. But you're right many states considered themself successor of Rome like the Holy Roman Empire. Also when the Western Roman Empire collapsed the Eastern half (which we call the Byzantine Empire) lived on for another 1000 years. They never stopped calling themself Roman even though they controlled no Italian territory. Even Russia tried taking some of Rome's shine by labeling themselves as successors through the Orthodox Church (which was the church of the byzantine empire) and by calling their leader Tsar (Ceasar). Note: Germany did this too with Kaiser.
More on this idea of a "Third Rome" can be found on this wiki article:
https://en.wikipedia.org/wiki/Third_Rome