Part of what's confusing about the wiki family page is that it's describing multiple methods -- and a fair bit of it is not up to date.
I have managed to make it work through this method:
https://www.mediawiki.org/wiki/Manual:Wiki_family#Multiple_wikis_sharing_common_resources
I can confirm that the list of things they say you can share is no longer accurate; I can't recall which of the folders isn't viable to share off hand, but at least one of them is no longer shareable by 1.30.x.
That may not be what you want, though; which option you go with is going to depend a lot on your end goals.
Are you looking to run one game with access to multiple wikis? If so, look at that option. (This is what I was doing when I was able to make this work -- a core wiki with the RPG data, a private staff wiki for issues and behind the scenes metaplot notes or dev, and a player-side wiki. If this is what you're doing, you or whoever is doing your setup probably also want to look at Interwiki to port data from one to the other easily.)
If you are looking to run multiple games that are completely independent from each other, potentially run by different people to provide general hosting, you or whoever is handling your server foo should take a look at the Zero to MUX thread; Nemesis describes means of setting up multiple users and handling the installs more distinctly that way -- and you would explicitly not want to use the 'shared resources' option described above, as your various game runners may want to use different extensions or combinations thereof (some of which have a habit of breaking or losing support between mediawiki updates).
You're less looking for a mediawiki person here, and more looking for a server admin. This is better news than you may think; more people are familiar with server foo than the weird minutiae and vagaries of mediawiki in particular.
Suggestion: list the extensions you want to use here. Some have additional requirements and dependencies, and these requirements may better inform your server admin how to handle your setup.