I was leading discussion for the course I TA for, and we were covering nationalism. I was explaining the 19th-century notion of nations and nation-states when a student asked if Rome would qualify. I did not answer because I basically do not know anything about anything before 1485 (I'm a modern British historian). All I could tell him was that I don't know if Rome would've qualified as a nation-state, but the term would be anachronistic.
There are certain aspects of the Roman Empire that can be thought of as very much like a nation state: a unified sense of culture, a common cultural narrative, and a feeling of kinship. However, who exactly these applied to was always somewhat vague, and usually tended to apply to Rome as an abstract concept more than any individual. And so everyone in Rome may have thought "I am a Roman" and "the Romans are descended from Romulus and Aeneas" this did not mean "I am descended from Romulus and Aeneas".
Perhaps more relevantly, the Romans did not view their empire as being the political form of the Roman people, rather their empire was that of "orbis terrarum" (the circle lands)". Not only was "Roman" a much more fluid category than, say, "French" but also non-Romans were not considered unusual or undesirable parts of the empire.