Did you install it through Galaxy or did you use the offline installer?

There is only one GOG Galaxy game which is GWENT and every other game can be played without Galaxy.

If i remember correctly, the last time i installed Cossacks 3 it required a CD key to be played in multiplayer

Yep. It is on the 'multiplayer games that require CD-keys' list: https://www.gog.com/forum/general/gog_games_with_serial_numbers/page1

THE LAST BLADE 2 uses Galaxy to manage its save file. You need to have the GalaxyCommunication service running for it to read and write the save. It affects unlockable gallery images, which are all shown as locked after restarting the game without Galaxy, even when you unlocked them before.

The ingame achievements list is also not accessible, showing a message that you need to be signed in to use "online features". I can accept that they unfortunately implemented the online multiplayer DRM'd or even the leaderboards, but there's not a single achievement tied to doing anything online.

To verify, I installed the game via setup on a system that never had Galaxy, and the game doesn't save there. Any gallery images I unlock are gone after I restart the game.

From a pure singleplayer gameplay perspective, it doesn't affect anything. It only matters if you care about having the images for the gallery/achievements in the game (the png assets are easily found in the installation folder).

I like using Galaxy, but I also like keeping saves from all games I play through and it was a pain to track this one down. In case anyone needs it, the game saves here: %LOCALAPPDATA%\GOG.com\Galaxy\Applications\53185476640680089\Gameplay\<userId>\gameplay.db

Just to make sure I understand correctly: if you install the offline game, saving the game works, but achievements aren't saved, correct?

No, if you install the game offline, it doesn't save at all. You just see your gallery progress for the current game session but lose everything once you quit it.

EDIT: And that is "offline" as in "without Galaxy". You can view the gallery without an online connection, however you need one to access "online features" and that includes the achievement list.

Galaxy (Online): The game is fully functional
Galaxy (Offline): The game saves but online features don't work
Setup (regardless if online or offline): The game doesn't save and online features don't work

You're still not making it clear if you're talking about saving game progress or achievements. I don't know what this "gallery progress" is that you're talking about.
The main question is: Does the game save game progress without Galaxy (or any internet connection).
edit: I just saw your last edit: so it needs extra software (but not an external service) to save game progress. I'd call that offline DRM....

Well, no reason that you consider valid anyway.
One reason would be that they don't want to bother to develop a authentification system and rely on existing services which PS Store, Steam or Galaxy provide instead.
I personally consider that a valid reasonl.
I am a programmer myself and I see no benefit in developing something that others already did and I can use for free or for a small fee.

Just to put things into perspective, the game is a port of a 20 years old arcade fighter game (i.e. Street fighter 2-like), it's not like there is really anything to save, there is no "save game" to speak of and the only thing to unlock that I am aware of is the gallery, which is just some low rez pictures of some character or scenery.

I tested, and while the game is fully playable offline without Galaxy installed, the gallery is apparently not saved (but all the others settings are). Personally I think it's more of a "bug" that nobody noticed because nobody cared about the gallery, IMHO it could be worthwhile to open a support ticket about that.
There is no real "progress" to save, the game is a basic arcade fighter game, start a new game, select a fighter, play until you reach the final boss or you run out of credit, (think Street Fighter 2), but after some fights you unlock one or more pictures that you can view in the "gallery" and that part is reseted once you quit the game.

My mistake then. I looked up the game and read:
"GAME FEATURES:
An epic storyline during the twilight period of the SAMURAI!"
That sounded like some kind of campaign to me...

Still: you say "reach the final boss" - isn't that a progress system? But OK, maybe there wasn't a save system in the original game either.

Yeah in the same way Street Fighter, Tekken and Soul Callibur are know for their "epic" stories ;-)

No it's really like SF2, for example : you chose Ryu, first you fight Ken, then you fight Chun-li, then you fight..., etc... then at the end you fight Mr Bison and if you win you have a small static ending "cinematic" depending of the character you chose and you are back at the beginning. Think 15-20 minutes to reach the final boss. (It was made to be played in the arcade)

Thanks for the explanation! (to me) it doesn't sound like that game belongs in this topic.

I would expect standard libraries already exist for such fundamental multiplayer features as LAN/direct IP connections. If not, they could (and should) certainly be developed. Sorry, but I don't see why developer convenience should be any justification for DRM.

Ah! Thank you for the clarification. I wasn't aware that the game doesn't save the game progress at all. But as an old arcade game, that makes sense.

So it's just the gallery, an ancient form of 'achievements'. I agree, tha this looks rather like a bug than an intentional DRM.

@Kevin: did you open a support ticket? And did they reply? It would be nice to know whether they are going to fix this or whether this is an intentional 'feature', that is going to stay.