I've often wondered if it was a universe where the Roman Empire never fell. I've always gotten a little bit of Roman vibe from the Empire.
Unlikely. More probable that the Terran Empire just did what most Western cultures did and adopted elements of Rome to paint itself as an inheritor thereof. (Just look at the architecture of any American or British government building...)
Besides, it's ethnocentric to assume that Rome was the only empire that mattered. The Han Dynasty in China was just as big, or bigger, at the same time that Rome thrived. And though many dynasties have risen and fallen since then, the case can be made that Chinese imperial culture persisted pretty much continuously up until the Communist revolution (and arguably the Chinese Communist Party is just the next imperial dynasty, for all its pretensions otherwise). And until modern times, I daresay Chinese civilization had a more widespread influence on human history than European civilization did. Indeed, much of modern European history is a response to Chinese civilization. The whole reason the Industrial Revolution happened was because Europe envied China's wealth and was motivated to devise new manufacturing methods to compete with Chinese goods and faster transportation methods to facilitate trade with the East. And the Enlightenment-era thinkers of Europe who invented modern democracy were motivated in part by Confucian teachings about how the responsibility of a ruler was to the well-being of the governed.
Besides, would it really have made that much difference to world history if the Roman Empire as a political entity had continued? If the premise is that it would have continued to expand until it conquered the world, that's ludicrous, because any empire that expands too far is doomed to collapse (a point "Mirror, Mirror" made explicitly), and of course there were empires in China, India, and elsewhere that would've had their own well-armed objections to the idea. So it probably couldn't have grown much more, just entrenched and stabilized while other empires and states developed much as before. Besides, by the time the Western Roman Empire fell, it had been thoroughly Christianized, and Christendom continued to be a dominant force in Europe for centuries thereafter, as well as an imperial force in its conflicts with the Islamic world such as the Reconquista and the Crusades. Heck, most European rulers painted themselves as successors to Rome, and Rome still had considerable political influence through the Pope. So I'm not sure the persistence of the literal imperial state would've had that much of a practical difference on the way history unfolded, either in Europe or certainly where the rest of the world was concerned.