While I'm sure the analogies could very well apply to other historical empires, and I assume my studies in Latin skew my objectivity, I see a lot of similarities between Westeros and the Roman empire in general.
I have not read the books (yet), so I can only speak to the TV show's similarities.
- They worship a pantheon of Gods, while still acknowledging an older pantheon. The Romans still acknowledged some people prayed to the Greek gods, even though they had their own version.
- The relation between Westeros and Dorne is very unique. They are still allowed to call their own rulers king/queen/prince/princess, even though that it not the case for the "regular" houses that bend the knee to the Iron Throne. This is very reminiscent of the Pax Romana, as the Romans left much of their new allies' government in place, with the exception left in place that Rome can intervene or ask for aid when needed.
- The free folk are analogous to the Gauls, in that they were separated tribes who only united to fight their common enemy (Westeros/Rome), and generally didn't get along too well. Julius Caesar specifically used his famous "divide et impera" (divide and conquer) to break this strenuous Gaulish alliance. Part of the reason some tribes yielded was to receive Roman aid through the Pax Romana. Some of them literally made peace because winter was coming. Many Roman encampments existed in Gaul, to keep the population in check. Due to the harsher region and the distance from Rome, criminals were given a second chance to join a legion in Gaul, much like how the Crows man Castle Black.
- The lands across the Narrow Sea very much resemble the North African regions during the time of the Roman empire. Very city-state like in government, and geographically very separated from one another. Also, there is a very notable city in those lands, whose rule is based on economic power and an counsel of rich men, with their armies mainly consisting of mercenaries paid for by the coin they make. For Rome, it was Carthage, for Westeros, it's Qarth (the similarity in the name is what gets me).
- Daenerys' campaign across the lands across the Narrow Sea is very similar to how both Julius Caesar and Marcus Antonius led their campaigns; in an attempt to solidify their claim to Rome. The only difference being Daenerys later decided to settle instead of returning with the newly recruited troops. Then again, both Caesar and Antonius also stayed in Egypt longer than expected due to Cleopatra.
- Nero burnt down Rome (although the specifics regarding this event are still widely discussed today), much like how the Mad King seemingly turned on his own people (and burned them) in the end. Both Nero and the Mad King have caused the downfall of their lineage as rulers. Both would after their demise be actively used as a warning for future rulers, showing how not to rule. This is actually somewhat analogous to our modern usage of using Hitler as an example of pure evil. It's just the most recent event of the most evil deeds we can think of (as it was for the Westerosi, and the Romans).
While your analogy of Caligula and Joffrey is somewhat correct, it would seems most of the similarity boils down to "young narcissist with a penchant for sadism". This can easily be attributed to how their enemies spoke of them (though we have seen Joffrey be exactly that), or simply their youth leading to naive leadership decisions which in turn led to dissent among the people.
Selfishness is a trait many teens experience some way or another, whether they're ruler of an empire or not. If anything, them wielding supreme power leads them to being more selfish as no one has the power to correct them on it.
I would add, though, that Tommen currently reminds me of Augustulus, the last Roman emperor. He was very young (roughly 12), and did not actually wield his monarchical power because he was too weak for it.
Augustulus was easily deposed by the invading armies, and I would expect a similar fate for Tommen should there be another invasion of King's Landing (just an educated guess, since I haven't read the books).