Not so close. Many people associate Greece and Rome, but the truth is that they were much less associated than people like to think. Yes, some emperors were quite hellenized and built their lifestyles around Greece, but that was widely discouraged by the senate and people of Rome. The Greeks were considered as effeminate and attached to depraved and immoral habits, and it was very difficult for them to ascend in important positions in government. The base of Rome are the etruscan and italic cultures, like it or not.And yet they weren't the same Greeks of the Peloponnesian Wars or Thermopylae, nor of Alexander and Pyrrhus. Their culture was essentially the fusion of the Greek culture that had been there before and the Romans who came into power in that region. The Romans themselves were very into Greece, Julius Caesar's last words were in Greek, does that make him not Roman?
And more besides, the only Latinness about the Westerners was the fact that they had adopted Latin artificially for their administration, it was adopted by them like the Seljuk Turks took on Persian culture, yet the Seljuks are always considered a Turkish dynasty.
Moreover it was rather convenient for the Pope that once Roman/Byzantine power had receded far enough away from him that he could curry favor with the new powers in the region by declaring their most powerful leader "Roman" Emperor when he had no authority to do so (the Senate had that power, or a reigning Emperor). It was nothing more than a convenient charade where the Pope could elevate himself above the Ruler of the World™ and the Franks could pretend they had the legitimacy and authority of Rome. Both got the benefit of getting an upgrade, and the Pope in particular has then been able to claim he is above every secular ruler in the world ever since (because he said so in 800, that's the basis of his authority).
Strictly on topic there's no way the Roman Empire as it was in the classical era could be represented in this game very accurately, or very much fun at all either. I think the same reasons are why we reasonably shouldn't expect to see China. One playable character at the head of the country, get deposed and game over, basically.
And yes, the Pope didn't had the authority to declare anyone emperor, but due to the disappearance of the western half of the empire, anyone powerful enough could to aspire to the title, and remember that Irene recognized the claim of Charlemagne.