It really isn't more complicated. These are mostly the excuses that 19th century Western European scholars used to deny the Eastern Romans their Romanness, because they wanted to place themselves as the inheritors of Rome.

First, The Roman Empire was always bilingual. Greek always held sway over the eastern half and Latin over the western half. Once the west is gone, of course you'll switch to primarily using the language that most use. The Irish didn't stop being Irish because they primarily speak English now, for example.

Second, the only states that refused to recognize the Eastern Roman Empire as the Roman Empire were the states that were descended from the German invaders and that was because they wanted to claim the Roman mantle for themselves. All of Islam called the East Romans, Rum. As did all of the rest of the world when they had contact with them.

Third, the switch to Christianity had already happened by the time of the split in 395.

Fourth, Rome hadn't been the seat of power for many Emperors since before the Third Century Crisis.

Fifth, states lose and gain territory all the time.

Read The New Roman Empire by Anthony Kaldellis. If you want to read more about this.