I can only do the theory here. A lot is dependent on how for instance GOG saves user data locally. And how cloud data is handled.
I believe GOG is more open than Steam. Steam does have a feature called family sharing but it has some limitations:
Game cloud data is part of the account where the game is played under. So all situations I see described on this thread assume a shared account for GOG:
But GOG is DRM free so games can be played offline. Keep that in mind when buying games. I expect you will have more options with GOG then with Steam in this matter.
On with the technical aspect. And disregarding the cloud part. And forgetting about Steam as its not DRM free.
I have Blade Runner installed. This one runs in ScummVM ( https://www.scummvm.org ). The save data is in this location:
Another one is Final DOOM. A dosbox game. The windows version running through wine. Here all save data, cloud or local, is saved in this location:
When running through GOG Galaxy these locations are most likely synchronized to the cloud. I run my games locally from Lutris.
So what I would to is create a clean install, zip it, and put this in a central location. You can create a bash script to extract it to a location in the users’ home directory. If you configure Lutris to look for GOG games in the users’ game directory then it should see them.
That’s the closest you can get without violating any rules.