I feel like we get questions like fairly frequently, but the best place I can point to is u/CrankyFederalist's answer from last year:

The Byzantine Empire was referred to simply as the "Roman Empire" during its time as a state. Did the average Roman care that they did not hold the city of Rome?

Very briefly, no, there was no contradiction at all. Status as a citizen was far more important than living in Rome, which was after all just a city. After the 4th/5th centuries it wasn't even the capital or economic centre of the western empire. Both the western and eastern halves of the empire were simply where Roman citizens lived.