They came from the Roman Empire though, that covered most of Europe. You couldn't call them Italians, as Italy didn't exist then. There were a lot of other city states though, such as Sparta, but similarly not all the Spartans came from there. You took on the 'nationality' of the people who conquered you.
The generic word 'Roman' applied not just to citizens of the city, but also to those of the state or the empire. In the same way, I'm sure not all Aztecs lived in the city of Tenochtitlan (Mexico City).
You didn't have to be from Rome to be a Roman - certain people and towns were granted Roman citizenship as a reward for loyal service (or bribe to stay quiescent). St Paul, for example, was a Roman citizen, because he came from Tarsus, which was granted citizenship rights.