The Roman Empire was never based on linguistics or national identity. In fact, national identity was alien to the Romans. Roman was who possessed certain rights, who observed the imperial cult and so on. Anybody who fulfilled these criteria would be considered a Roman. Emperor Karl the Great was not a Roman in a linguistic and national sense, which would not have made sense to anybody back then. But we must not forget that many Roman Emperors were Syrians, Gauls, Dacians, Thracians and some were Germanic like Karl.
I recall watching a series on TV and watching as a young German professor told the audience about how she came to devote herself to becoming a Medievalist. She related how she was watching something years before as a youngster on German TV. The program ended with Karl der Grosse (Carolus Magnus) being crowned by Pope Leo on X-mas Day, 800 and the announcer stating that "and the Middle Ages began". A somewhat different perspective. For those interested in the linguistics of Charlemagne's empire, look up the "Strasbourg Oath" of 843 which was written in Latin, Frankish German, and a proto dialect of French.
I am nor certain of this but I think there was something to the notion that if you were or became a Roman one of the things you did, really had to do , was to learn a non barbarian language to be considered a good Roman. For the Romans, language was considered the vehicle of culture. Either Latin or Greek would do. When getting to the barbarian invasions and the fall of Rome (in the West) I would joke with my students that I was probably the only person in the classroom who would not be considered a barbarian in Classical terms because of the linguistic requirements to be considered.
The crowning of Charlemagne on Christmas Day in 800 was in part a reaction to the Byzantine ruler Irene. Until that time the West had looked up to Constantinople for leadership in culture and monetary system (the solidus). The early Germanic rulers of the West claimed to rule in the name of the emperor in Constantinople. Excerpt from Wikipedia's Irene entry: Irene's alleged unprecedented status as a female ruler of the Roman Empire (there had been women ruling the empire in their own right before, notably Pulcheria) led Pope Leo III to proclaim Charlemagne emperor of the Holy Roman Empire on Christmas Day of 800 under the pretext that a woman could not rule and so the throne of the Roman Empire was actually vacant.