Why couldn't most of the Mediterranean basin come to think of itself as Roman? It would do wonders to solve the problems of nationalism, and it would give imperial aspirants a powerful incentive to pursue empire during the inevitable "warring states" periods.
King Gorilla
A lot of them did. You see discussion of the bitter hated and hostility between 'Romans' and Germans, the former being defined as all members of the empire. [Albeit this was partly due to the religious differences between the assorted Roman sects and the Arian Germans at a point when the external threats and recent final [formal] destruction of preceding religions had left sectarian feeling very high.
Similarly a lot of the various Germanic rulers that emerged not only wanted initially to be allies of the empire but later to seek to reform it. The idea of empire maintained a strong presence in the Mediterranean [later European mind - don't forget Moscow's claim of being the 3rd Rome].
Steve