By the game description I don't see any Galaxy achievements. Does anyone know if they will be added in a future update? I'm moving away from Steam but I love achievement hunting and want to pick this game up on GOG. Just need the addiction of achievements!

Same here:
Moving away from Steam, and ot GOG, but missing Workshop and achievements, painfully...

Any news ?

asking in partially related thread cuz I cant create my own from current browser
Does anybody know if daily runs work without galaxy? (for example - dead cells require this completely optional client for such feature)

I haven't tried them for this game so I don't know, but when it comes to Dead Cells doing the daily runs only requires having an internet connection. You only need Galaxy if you want to register your score on the leaderboard.

only for windows version, I think. Since the linux build cant play daily runs due to unavailability of galaxy:
https://www.gog.com/forum/dead_cells/daily_challenges_on_linux_without_galaxy

That's annoying, all version should be equally playable. It's a shame that this hasn't been fixed yet.

As for Slay the Spire, I've given the Daily Challenges a try now and I assume, but can't guarantee, that they won't be a problem for you, as I'm using the Windows version and so don't know how it plays on Linux. However, It would surprise me if it didn't work because the daily challenges specifically have an off-line mode, so internet related issues don't apply. there.

To also answer the original question, while I don't use and therefore don't know about Galaxy achievements, there is a statistics menu in-game where achievements are tracked, in addition to other stats on how well you're doing.
There's also a record of every session you've ever played, which includes the seed code so that you can replay that layout if you wanted to.