I've spent a lot of time (on walks) puzzling over what exactly the Old World laws are supposed to be. A big chunk of what I get out of mapgaming is basically interpreting the kind of abstract game into a full, rich story. And of course Old World doesn't really have the flexibility or detail to portray a lot of government forms. What I've come up with:
TYRANNY VS CONSTITUTION
The basic axis about the meta-rules about how political rules are made. This says nothing about whether there is a well-organized body of law itself, but is instead about the notion that there is law about how you make and interpret law. So a constitutional society can be completely despotic or republican, it can have a limited or a completely unlimited government, but either way it has clear procedures about how its own government is constituted. Tyranny is pure despotism.
DIVINE RULE VS LEGAL CODE
The underlying ideological rationale for the state, as opposed to its meta-rules or its actual laws. A Legal Code society roots its authority in the notion of an underlying (whether changeable or unchangeable) legal code. It may or may not have aspects of divine rule through its state religion, but the existence of that particular state is justified by it being the representative of a law code representative of that culture. Rome, for example, had pagan state religion (something that contradicts a legal code in the game rules), but the Roman identity was ultimately rooted more in its civil law than in its religious institutions.
Divine Rule, on the other hand, represents a society going higher than just a cultural identity/set of practices to claim direct authority based on some religious rationale. Not just that it is there, but that religion, not cultural tradition, is fundamental to the state. This is why a pagan religion can only be a state religion through Divine Rule: if you practice a religion that is peculiar to your culture, that does not make the sort of universal claims or the demands of intense and exclusive loyalty of a world religion, it is a weak foundation for power. Divine Rule could be represented in monarchical or republican fashion, it can be tyrannical or constitutional, but it does mean that the underlying ideological basis for the entire society is its choice of religion, not its own people.
World religion Divine Rule likewise takes on different flavors depending on whether you control the religious head/holy city or not. This can be God's Holy Governor like a Pope, a Caesar like the Byzantines, a Caliph, but it can also be like a feudal society where England and France and Spain are all distinct cultures and states but are merely subordinate to the religious head in their need for authority.
Combine these with other laws and you start to get a feeling/can tell a story. I do think the game could really use Polis as a third option in between Centralization and Vassalage, though. I usually roleplay Vassalage for Greek and Mesopotamian style city-state civilizations, but to me "Vassalage" really implies Persia (devolved government) and Carthage (vassal swarms).
MONOTHEISM VS POLYTHEISM
This really really should be a Doctrine for World Religions, not a law, and I hate its premise. What I see this as really representing is a distinction between centralization and decentralization in the religious institutions, since Polytheism is necessary to propagate shrines (organized folk religion) beyond a few sites. This got me thinking about how religions really spawned. I said I like this better than Civ VI's Mad Libs religions, and I do, but I do think there is something off in having Judaism spawn from Ranchers. Like, yeah, having a lot of shepherds is certainly very Jewish, but I'm not convinced that shepherds are the thing that caused Judaism to evolve...
To me, it looks like historically monotheism was developed generally as an attempt to streamline the religious institutions in order to better control them. (Really, to make them practical state religions.) Akhenaten was motivated by that; the Hebrews merged YHWH with El to create the Jewish God; Zoroastrianism did that with its Aryan paganism.
What makes Judaism different from Zoroastrianism, though? You could certainly argue that Judaism may have been spurred on by the Zoroastrian example, and definitely that Christianity has an imprint of Zoroastrianism on it through its doctrine of a messiah. But I think you could also just as well argue that while we might be able to trace the idea, historically, it isn't necessary; you can have no Zoroastrianism in the world at all and still independently come up with Jesus, whereas you can't have a Jesus without having a Moses. Difference between being influenced by ideas and literally building upon ideas.
To me, I think what it comes down to is that Judaism's character is defined by being an ethnic sect in a highly competitive world. The Jews were in a deathmatch with other Phoenician cities - very decentralized - and developed their religion as an expression of cultural narcissism, an adaptation that helped them to survive. Chosen People. On the other hand, Zoroastrianism became a tool of universal monarchy since it spread with a culture that was quickly conquering others. Finally, Christianity as we know it is arguably as much about having other cultures, even another universal monarchy, interact with Judaism as it is the metaphysics. Christianity requiring a twelve-citizen empire isn't that bad, since it can kind of kill two birds with one stone by representing the role of the urban poor in propagating it and the fact that the realm is big enough to have that many urban poor (and hence a reasonable candidate for universal monarchy).
I don't know how exactly you represent these things, but I feel like it would involve choices of laws, neighbor interactions, whether or not you have diverse cities.
ROLEPLAYING
For example, as Egypt I recently had a game where I wound up being forced to convert to Zoroastrianism. I was on Highlands and Carthage forward settled me. I was blobbing fast, and I wound up in a war of choice through an event, but Carthage stomped me. Later THEY provoked another war and were screaming down on me, with a push would have overrun the whole empire. But I brokered a treaty in time, was subjected to Versailles-style reparations. Had to basically set everything else aside and run nothing but Treasury projects and Hamlet construction, massive economic growth, to stabilize, although it laid the foundations for a massive boom.
Normally I would roleplay switching to my homegrown religion (Judaism), but I couldn't do anything but try to realign diplomatically with Zoroastrian Carthage, and so I did. In time I outgrew them (they were boxed into a corner) and ultimately consolidated my victory (I was heading to win anyways) by overruning a very, very weak Greece that had already been broken by Assyria. My empire didn't totally span the map from west to east, but it nearly did, I had a huge frontage against the two Manichean Mesopotamian realms to the south, and had become the only empire to really punch into another nation's plateau.
So the way I ended up interpreting it was that my Egyptian state was a Chinese-style bureaucratic monarchy. It was Tyrannical with a Legal Code, maybe Elites (I think?) and Calligraphy (writing is highly valued socially). Judaism was a popular, indigenous faith of the masses, while Zoroastrian remained a lofty and elite religion of the nobility precisely because Judaism is ethnic in character (YHWH's covenant with Egypt) while Zoroastrianism's tendency towards universal monarchy encouraged the deepening of relations with Carthage, Rome and Persia and because its ceremonial character mixed well with court pomp.
As a divided society, Egypt practiced Tolerance and Polytheism, yet also had Volunteer Holy War crusader armies. As Egypt emerged onto a global stage, the equivalent of any two of its three coreligionists, the preeminent world power, centrally-located and along a major religious faultline, it began to develop a notion of Zoroastrianism as a sort of religion that was compatible with and could contain/tolerate other religious traditions while still being of severe importance in and of itself. Religion that doesn't have to kill other religions to spread itself, but does insist on itself. Carthage became the old civilization that controlled the clergy, but Egypt became the political-diplomatic-military face of it, much like France, Spain, Austria or Holy Rome as a Defender of the Faith even if the Papal States was the intellectual core of Catholicism.
Egypt was also a fairly free society (most of the "free" policies just look clearly better to me, like Colonies and Freedom and Vassalage if there's not a ton of marshes around).