Religion often gets the blame for violence and war in human history, and it's fair to say that it should accept responsibility for some of that. I think, though, that in the West we isolate religion too much and try to assign blame on that single factor, whereas religion doesn't stand alone at any point or for any person. Or to put it another way, religion can't be added-in or subtracted-from in the same way that a sport or hobby can. So what does a religion do?
Religion gives identity, and in most cases it adds to identity.
Throughout human history, identity has been a combination of three factors: religion, land and ethnicity. Even though we can break down these into smaller categories, the big three are religion, land and ethnicity. When we read about any population, we always read about these three. The key to understanding how religion fits is to first understand that these three factors can't easily be pulled apart. For example, the Babylonian people (ethnicity) lived in and around Babylon (land) and worshipped Babylonian gods (religion). They understood that the people in the nearby country were a different people, lived in a different land, and had local gods.
The three elements together provide the binding features for the society and if any of those factors are threatened or removed, it causes a problem. And in story games, problems make story.
In your game, you can use religion to give your characters identity by tying it to the land and the people. Lore checks can reveal that the Lothir people in the Handar Valley worship Ahud the sun god, for example.
Individual characters can also gain identity from the religion. In Christianity, St Paul was Saul before he became a Christian. The conversion moment was enough to warrant a name change. Even today, converts in some countries change their names to one more suited to the new religion. Furthermore, children born into a religion are often given names of religious heroes, named by parents to honour the antecedent and perhaps as a hope for the life of the child.
Identity can also come through religious observances. A religion might have a systematic marking system in either clothing or tattoos. Conversely, they might abstain from tattoos altogether, or engage in some other flesh alteration like circumcision or ritual piercings. Consider also how hair is worn or shaped. Is there a religious significance to it? What happens to the Samson character who has long hair as religious devotion, and then has it forcibly cut off?
Religious symbols also provide identity markers for peoples and characters. It might be incorporated into the battle standard for the people, or on the currency ("In God We Trust").
The dark side of any such symbols is the obvious definition of who is an insider and who is an outsider. In Nazi Germany, homosexuals were forced to wear pink triangles on their clothes, and Jews had to wear a yellow star of David. The presence or absence of symbols in a society can be telling for prejudices or honour, and compounded further if those symbols have social and religious meaning.
Religion provides identity in many ways. In your game, you can use this as background: descriptions of architecture or idioms. You could use it as deeply felt personhood, in which a people group believe they are chosen by the gods for a special purpose. You could run a story about the ostracism associated with conversion and apostasy. Religion and identity are tightly woven together, and can give depth to your characters, and in turn provide plenty of drive for stories.