So for coding, I know about version control, Git, all that good jazz. And for artists and musicians, they can work on individual art assets and songs.

When there's only one writer, writing's a lot easier.
But for larger teams, how do you collaborate?

Are people assigned certain characters to write for? How do you avoid discontinuity? How do you control versions, does everyone write to the same script?
I can imagine people being assigned to write for different parts, but how do you prevent characters from sounding different when a different writer does the dialogue?

Or are the number of writers in a game kept to a minimum to prevent these troubles?

Sorry for all the questions, feel free to ignore them since they're basically restating the same question differently.

I don't have any game industry experience, but I have had collaborations with non-game projects.

I think that the approach you're describing is a bit too fragmented and independant to work for collaborative writing for the reasons you described.

I would think that you want someone in charge of the writing team who can assign tasks appropriately and return them if necessary. Writers are given things to write, but they don't need to work in solitude; if there's a continuity question, the writer should ask rather than guess and produce unusable copy. If there's any issue with what's been written, then it's up to the person in charge to notice and have it fixed.

Regular group writing sessions can be useful because a lot of people in reviewing work and in planning future writing, so everyone's on the same page (sorry for the pun). Templates and guidelines can help as well in maintaining "voice" of characters.

I think it's similar to hand-drawn animation. The artisis have models to reference and specific purposes for each assignment they're given. Then there's a director (or some similar role) who reviews the work and makes sure it fits together. Any contribution which doesn't fit correctly is re-done.

I think there are topics that can be accessible to anyone, as example minor characters, can be used by anyone. While leading characters are in the hands of the lead writer only.

small tasks can be easily assigned and the results included in the main plot with no problems. Anyway, the big work should be of one's mind only. This is how works for cooperative writing.