This, so much this. Being Roman didn't mean you spoke Latin-half of the empire's citizens were Greek culturally, and spoke greek. Since the time of Augustus at least, Roman officials in the east were required to be fluent in Greek. Being Roman was, much as you say, the same as being American: it was more about the identity than any unifying culture.
As already mentioned, the Rhomanoi living in the ERE were fully convinced to be romans and living in the roman empire. There is no doubt, that from the roman point of view, the roman empire existed until 1453.
Actually I already confirmed 1453 above. But this is again fully irrelevant to me!
The romans did not think in epochs and they did not ask this weird question. It is a question of modern historians, which tried to divide history into epochs. One important reason for such epochs was the question: when did the ancient age end and the mid-age start? Combined with this question they combined the question, until when can we call the ERE a truly roman ancient empire, and when did it become a mid-age kingdom called Byzantine Empire.
Therefore we have to define, what ancient age means. And what defines a roman empire as distinct of a byzantine empire. Also the really tricky question: What does "Fall" mean? Of course from the point of view of a modern historian. The opinion of the romans of their times is not really relevant.
Remember, the initial question in the title of this thread was not: When did the sovereignity of the roman state end?
The question was: When did the Roman Empire fall?
And this question is a fully different beast. The answer to this question is unknown since 1500 years and fills libraries.