Using your first linked quest as an example,
Ahhh! I'm sorry,
<If(PlayerParameter(4))>m-m-madam<Else/>s-s-sir</If>! No need to get angry, now! I-I have a problem, all right!? I was working it out with my mummy, and─uh, er, ah...
It's quite literally a template and placeholder system that generates the correct line based off your character's information. Just like any other templating system out there, you can have conditionals and branching which allows for a more "dynamic" text block display, which is what the game already does. Making an adjustment to check for some other character variable would be simple and like all other question dialog, handled completely client-side because the server doesn't care about pronouns. Naturally, the character's gender would still matter for things like equipping gear, but those checks do have to be validated by the server. There's no in-game dialog that checks the gender of your party members to my knowledge.
The plugin itself basically just overrides those local to the client variables and the text payload continues as normal. It'd be a lot more involved to inject something like "they/them" since that's not swapping a conditional if/else, but the general work needed to make these adjustments could be largely automated by the developers, I'd imagine as their scripting system likely allows them to already make those changes without having to hand-touch each one. (Just like how all the quests that mention race magically adjusted to Hrothgar and Viera).