...Or is it required? What about multiplayer?

EDIT: Never mind, the store page was updated to answer the important part: Galaxy is required for Daily Challenge Runs. (According to another user answering me elsewhere, multiplayer is local co-op only, so Galaxy presumably is not required for that.)

Ya, it seems co-op is local only. For some reason I thought this game worked online but I must be confusing it with another game...

I just tested it again, the daily door was locked with it's timer on it (Because I went in it to test it a few minutes ago).
The leader boards loaded as well when I opened that menu. I also do not have GOG Galaxy running, so you do not need it to do the dailies and view the leader boards as far as I can tell.

It's a fun little game though, I highly recommend it.

You don't need the client itself running, but the game does need some elements that Galaxy provides access to. Without the client installed, clicking "Leaderboards" will just display "Not connected to GOG", and the Daily Run door will always be closed. I mostly agree. Despite the frequent comparisons to Spelunky, the game plays very differently. I doubt I'll ever get into it as much as I did Spelunky (or get that good at it), but it's still pretty decent.
I just wish the ranged attacks worked more like in Terraria, where if you're using mouse and keyboard, aiming is done with the mouse cursor. Numpad aiming sucks -- and frankly, I don't like being limited to 8-direction aiming in general, no matter what control input method is being used.

Dang, good investigation though. I didn't think of the game using things Galaxy installed.
I've been tempted to uninstall Galaxy already because it's been deleting my game shortcuts and replacing them with shortcuts to itself.