Roman could mean multiple things. In one sense, it could simply mean those from the city of Rome. But Rome was unique in the ancient world (at least I'm comparison to Greek cities), from its earliest history, in being willing to give citizenship or partial citizenship to many of those it conquered. Rome also uniquely granted citizenship to freed slaves. By the end of the first quarter of the first century B.C., all Italians had Roman citizenship. A man like Cicero, born at Arpinum and the first in his family to become a Roman senator, considered himself a Roman and a man of Arpinum- he claimed to have two countries.

And by his time, citizenship was starting to be distributed to individuals from the provinces: Greeks, Spaniards, and some from southern Gaul. We have speeches from Cicero dealing with issues that arose from these citizenship grants (the pro Archia and pro Balbo) and even an inscription recording the decision of a military commander, Pompeius Strabo, to give citizenship to Spanish cavalrymen. As time went on, it became more and more common for provincials, and especially elites to gain Roman citizenship (think of St. Paul, who made that claim). And in 212 AD, the emperor Caracalla made all citizens of the empire citizens. Everyone could claim to be a Roman.

But whether these citizens actually identified as Roman is hard to measure, and the process of provinces "becoming Roman" was long and quite different in different parts of the empire. Some could identify as Romans when interacting with other Romans, while also embracing local identities at the same time, as multi-lingual funerary inscriptions from North Africa show. Meanwhile, Romans from Italy or who had had citizenship for generations were often xenophobic, and particularly wary of freedmen. The satirist Juvenal, writing in the late first to early second century AD, is evidence for the types of prejudices against those who had come to Rome from abroad, even if they were Roman citizens. But, over time (particularly from the third century AD on), the city of Rome itself became only symbolically important, and more and more provincials identified themselves as Roman, without controversy.

This answer is just scratching the surface. It really is a very complex topic. For further reading focusing on Gaul specifically, check out Greg Woolf's Becoming Roman. it's also worth noting there's a lot of debate about whether we should use the term "Romanization" to talk about the process of non-Romans becoming culturally Roman. I personally think that term is flawed, but there is disagreement on that.