Be careful not to project the modern meanings and connotations of "empire," "emperor," and "republic" back into ancient times. I know more about the Byzantine era but I'll touch upon the Roman beginnings, too.

The concept "res publica" means something like "the public sphere/matter" ie the government, state, etc. A literal translation into English is the word "commonwealth" where you can see the etymological clues much clearer. Nowadays "republic" implies a representative character vs. monarchical, but in Roman times it did not, simply that the needs of the public were somehow addressed by the government vs. a despotic, tyrannical rule. "Imperium" was the most supreme sort of authority in Roman political theory and "Romanum" is the adjective "Roman" in Latin. So think of Imperium Romanum as "the Roman Authority" vs. a Roman state ruled by an emperor. The anti-monarchy sentiments of the Romans prevented them from recognizing that their emperor was indeed a monarch.

In the Greek-speaking east, though, there were no cultural obstacles and they generally called it like it was. When consuls were the supreme authority in Rome, they were called "hypatos" in Greek which simply means "the greatest/supreme-est [person]." And when the figure of the emperor appeared, the Greeks called him a basileus, the Greek word for king. And what did a basileus rule? Much like a King rules a Kingdom in English, a basileus rules a basileia in Greek! So think of Basileia tōn Rōmaiōn as "emperor-dom of the Romans" and it makes more sense.

Purely by accident of history and the Hellenization of Rome into medieval Byzantium did Romans become reconciled to the idea of monarchical language. But that took centuries. And even then, there was the idea that Emperors ruled for the benefit of the empire as a whole, not as hereditary despots.