Of course "The Centauri Republic" is an English translation of the Centauri name.
And obviously calling the Centauri ruler an emperor is either an English translation of unknown Centauri title(s) or else English speakers use emperor regardless of how accurately it translates the Centauri title, merely because English speakers classify him as an emperor equivalent.
In one episode it was said that the Centauri Republic was founded by the first Emperor.
So we would expect that the usual English language names "Centauri Republic" and "Centauri Emperor" were chosen by persons to whom it did not seem as paradoxical as it does to SOME, repeat SOME, members of Science Fiction and Fantasy stack Exchange.
What was the title of Napoleon Bonaparte from 1804 to 1807?
Napoléon, par la grâce de Dieu et les constitutions de la République, Empereur des Français
I may point out that the Roman realm could be called an empire in some sense even while the republican form of government still functioned, between 200 and 100 BC, for example. The English word empire is based on the Latin word imperium meaning authority for military, political, and judicial command.
When Rome ruled most of the Mediterranean region the Romans began to claim that they had been granted corporate imperium (or authority to command and rule) over the world by the gods. Thus the Roman realm could be called the Roman imperium over the world and over lesser men, which could be translated as the Roman Empire, even while the Roman Republic was still going strong.
And the Roman Republic continued to function for decades and centuries after the usual foundation date of the Roman Empire. One could say that the senate and the popular assemblies merely changed their voting habits to almost always vote in favor of the policies and candidates endorsed by the Emperor. The legal fiction which was gradually abandoned over centuries was that the Emperor was merely a senator with extremely great personal influence who was granted a number of ordinary and extraordinary powers, titles, honors, and positions, so that he was a super duper magistrate of the Republic.
The English word emperor is derived from the Latin imperator which basically meant a magistrate with imperium. Since the emperor was a super magistrate with more imperium than all the others combined, he could be called THE IMPERATOR, so it makes sense that "emperor", derived from imperator, is the English word for an emperor.
Furthermore, for centuries the only person in Europe called an emperor was the Holy Roman Emperor, who was thus often simply called the emperor. And in those same centuries the elected emperor had very little political power over territories within the empire that he was not the hereditary ruler of. So the empire functioned more like a federal republic with a very weak central government and a president who had the title of emperor. The later Holy Roman Emperors may have been more similar to heads of state of a republic than monarchs, just as the first Roman emperors were in theory republican magistrates with extraordinary powers. The title of emperor has usually been both far more exalted than king, and also less like a monarch than king.
Since the Roman Emperors claimed to be the rightful rulers of the whole world, and since the latest Roman Emperors were a lot like presidents of a very weak republic, one could combine those concepts and imagine a government with very loose and weak authority over the whole world which functions as a republic but whose president has the title of emperor to show that he is the overlord of the whole world.
And if you think about that concept for a while you will realize that at the present the Secretary General of the United Nations is vaguely similar to an emperor of Earth.
We might imagine that the Centauri colonized many planets and that most or all of the colonies eventually became independent and sometimes fought wars among themselves. And eventually the Centauri worlds decided to form a federal union which was stronger than the United Nations but probably much weaker than the United States.
And the president of that Centauri Union or Republic took a title meaning something like "All-ruler", "Ruler of Everywhere", "Master of All", "Centauricrat", etc. to show that he was the leader of the federal union of all the Centauri worlds. And because that title claimed so wide-spread rule it was translated into English as emperor since one of the main implications of emperor is ruler of everywhere.
Of course the misdeeds of Emperor Cartega show that the central government was now much stronger and the power of the emperor much greater than I suggest was originally the case. We can suppose that just as in US history the power of the central government grew compared to that of the states or planets, and the power of the president/emperor to control the Federal government also grew.