If I don't have an internet connection and I launch GOG Galaxy and then Frostpunk 2, GOG won't let me continue from my previous session because the game doesn't see the saves. I can only start from the beginning, but I can't continue.

If I run GOG Galaxy without internet, it doesn't show my installed games (the sidebar is empty). I don't understand how you can start the game. However, I just started Frostpunk 2 from its executable without GOG Galaxy and without internet. Saves are visible. Try this way.

Have you checked the "keep me logged in" option in your settings? I have something like that and I can see my games when I'm offline.
I know I can continue my story without GOG, but I would like to have my stats and achievements, and that means I have to have GOG Galaxy and with it on FP2 cannot see my saves. You have to close GOG Galaxy to play FP2 offline. That means GOG prevents me from continuing my story when I can't connect to the internet.

The check is enabled. I think it's only to save your credentials and autologin when you open the client without requesting them. I disabled my internet card from Windows settings (simulating no internet connections) and GOG Galaxy doesn't load anything. Maybe it's another problem (version 2.0.77.22 Beta). However, try to go on the game page (installed, not store), click on the Settings (on top, next to Play button) -> Manage installation -> Configure, then go to Features tab and check the cloud settings.

EDIT: the Day1 patch seems to resolve this issue (source: Steam update changelog, Other section)
https://store.steampowered.com/news/app/1601580/view/4691151273074113074

This is most definitely NOT fixed for me. I have been unable to load games since the game came out unless I close Galaxy. I run in a quasi-offline mode, where my firewall blocks most programs from the internet; I will not change this for Frost Punk 2., however, GoG Galaxy is allowed through. Unfortunately, Galaxy doesn't handle this well. Certain features like play time tracking pass through, but not others, like achievement tracking do not. I don't know if this is a Frost Punk issue, or a Galaxy issue, but saves will not list while Galaxy is running.