If I were you, I would get something like Heroic to manage your games instead
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I've dabbled with Lutris and Heroic. But Lutris only showed me the Windows patches during its update (which obviously don't work).
Heroic on the other hand doesn't work as well as I'd hoped. All operations take a really long time and some of my GOG games don't show at all (in Lutris they're appearing). I'm using the current version 2.13.0.
And even if it works, I'd be interested how Heroic implements its update mechanism for Linux native games since it should have the same issue regarding available data :)
As save-games and configs of my GOG Linux games are normally(!) outside of the installation folders (found them in my home-folder, under .local or .config), I simply install such new version .sh file into the same locations (overwriting the existing installation). But I do not know for sure, if this is the best solution in all cases.
Anyway you can easily try it by yourself after backing up your game-folder (simply zip it). This way you will not lose anything if overwriting game files will mess something up.
EDIT: maybe it is worth finding the save-game/config folder and back it up, too. Just in case the new version messes with config files.
I considered just installing the new .sh file in the same location, but wasn't sure whether that breaks my saves in the worst case. So it's reassuring to hear that others had success with this method. I'll try it when I get an update for one of my Linux native games. Thank you :)