Nonbinary person speaking here.
(BAD IDEAS INCOMING)
If you’re really set on your world being European medieval times-based and cis-centered and still have nb in it, then you can make an option for what we actually do in real life if we live in places that don’t accept non-binary identifications (culturally, my country doesn’t, we’re very accepting of trans and queer folks, but nb? talk to people about it and they’ll look at you very weirdly)…
Give the MC an option to identify as one in their head, despite what the rest of the world see them as, despite what pronouns the rest of the world use for them. (For example, “I’m a man”, "I’m a woman’, and “I’m seen as a [gender], but I’ve always thought of myself as neither, and found it odd how people are so stuck on that part of myself.” That way, you can keep them in the closet and still have people react to them in a binary way, if you want. I’m not saying that this is a palatable idea for many nb folks, but it’s a way to do it with less coding hassles.
And if you want it to affect what they do, well…it can affect a lot? Being nb is as integral to us as being a man/woman is to cis people. The sky’s the limit. They can be very confused about what they are, if there are no role models in society. They might think they’re crazy. If the MC tells their party members, it can have as much of an impact as other personal things they tell them. A character in an insular society that never heard of such a thing before might be very puzzled. They might try to figure it out with the MC (trust me, being nb can be really confusing). Or they might, as a number of assholes in the real world do, accuse the MC of being off their rocker and lead to an inner-party dispute about their sanity. However! Since that is not fun for everyone (I enjoy overcoming a well-written asshole, but not all people do) you can have the MC struggling with UNDERSTANDING why people are so insistent on gender and how gender matters. You can have them not UNDERSTANDING how they are seen as their physical gender, not fully. That’s more important to the nb experience than just chafing under gender roles. And there are NB beings in the game, correct? You can have them identify more with beings who are, and have closer rapport with or wish to be like them.
And lastly : if the game is fantasy-ish, consider incorporating elements from culture that aren’t necessarily Popular European Medieval? It’s your world, if there are ELVES in it, there’s no reason why it should need to follow the standard model to the letter. History is more diverse than you think. If THAT is what you want, go for it, but there isn’t just any one way if you truly want nb.
EDIT : Disclosure, while I’ve played nb often in these games, I rarely play as ‘myself’, so a lot of these might be…invalid for people who do prefer to play as themselves to enjoy the story.