This game is advertised as multiplayer/coop, but there is no mention of it in the game itself. As there is no documentation whatsoever. Am I missing something obvious or is there a misunderstanding about the term "multiplayer" and in this case it means "links to social networks are included"?

I haven't tried it myself, yet, but as far as I understand it, it works like this:
To start a multiplayer game the player has to improve the stockpile in the initial world and must have at least 5 dwarves.
After the enhancement of the stockpile, the player can apply a special spell and choose multiplayer options.

I'm not sure whether the game needs GOG Galaxy for multiplayer or not, but quite some new releases do. So if it doesn't work like described above, it might be because you aren't using Galaxy.

I hope we can get an answer to exactly how this works. DeMignon's answer is appreciated but I'd really like to hear something from the developer or someone who has tried the game.

I'm on the fence about buying but no way I'm going to use Galaxy, the game would need direct connect or LAN mode.

That's kind of a weird convoluted and annoying way to do it, but not GOG's fault. Thanks for the answer HaaYaargh (Vanuatu? really??)

Multiplayer is also in alpha so hopefully it won't be so convoluted in the future.

So this is yet another game which does not have any modes of DRM-Free multiplayer. I'm not sure why when only a PART of the game is without DRM then that means the game is DRM-Free. At least GOG is finally admitting Galaxy is DRM encumbered.

I think you're mislead. I hate DRM as much as many other Goglodytes, but what you're describing isn't genuine DRM. Just because a client is involved, it doesn't mean it's Digital Rights Management. Sure, that's how Steam works in many cases: No login to your Steam account, no gaming :-(
However, the Steam client also offers some functionality for developers to make their life easier and save them development costs, e.g. a framework for easy multiplayer integration. Since the client is mandatory, this infrastructure is widely used to cut costs and development time, by simple refraining from implementing an own multiplayer solutions. In the past, the downside for GOG was, that those game's multiplayer mode couldn't be transferred to GOG, even if the game was DRM free, because of the missing Steam framework. So GOG offered Galaxy with similar functionalities to help developers releasing their games here, without the need of implementing multiplayer or achievements from scratch.

If you want to blame somebody, blame the game developing studios focussed on Steam and its framework. GOG is just trying to work the market and to break Steam's clever plan of having exclusive games due to offering an helpful but proprietary framework.

Btw, you might even be able to add a game locally installed to Galaxy and use multiplayer, even if the account logged in via Galaxy doesn't own the game on GOG. This should also be legal under certain circumstances, because GOG allows you to install the game multiple times on PCs in your household. If for example your child has an own GOG account and you installed the game on the family PC, your child could add it to their Galaxy session and play it without owning it themselves. Though, I've never tried something like this, because there's no such constellation at my home. Still, if it works, it'd be the definite proof that Galaxy is not about DRM.

There is something missing from the GOG CtW page.

"This game requires GOG Galaxy to play multi player".

Really something that needs sticking on there.

As for how to start multi player, yeaaaah, that is really screwy ;p

It is genuine DRM (sounds like something from Microsoft) because I can't play multiplayer without logging into a remote service to ask permission. If GOG goes away or my account is deleted or I don't have internet access then I can't play multiplayer anymore, even over my LAN or via direct connect. Sure there are worse DRM schemes out there but Galaxy is still DRM for these reasons. To be DRM-Free there must be an option that doesn't require 3rd party authorization.

So far as comparing it to Steam, GOG is still better because you can reinstall without asking permission. I'm even told that most non triple-a games you can run without starting Steam but I don't use it so I can't verify.

So the game is NOT multiplayer on Linux on GOG. Thank you so much for this clarification! GOG really really should state those vital information very clearly in each game page. I was wondering if i should buy the game on Steam or here, as i prefer DRM Free versions, but obviously only if the DRM-Free version is feature-complete.

GOG, i would like to buy more games on your store, but we (linux users) can not do so while you keep ignoring us. As said here, Galaxy seem to be required to enjoy some multiplayer games. Port Galaxy to Linux.