It doesn't even need to be via galaxy. I use heroic games launcher, and logging into gogs site still shows all my recent achievements.

I think the only pre-requisite is that you install via their API rather than the offline installers they offer. (Which both galaxy and heroic do, I'm assuming lutris too? Tho haven't checked it personally).

It's been a while since I used their app but I vaguely remember some support (?). Although at that point you're basically using GoG as DRM, but I appreciate the attempt to give Steam competition in that space. The whole concept of meta gaming in general is kind of monopolized.