The game page has a note that says:
Multiplayer notice - read here: Support
But clicking on it doesn't provide any relevant information

As far as I know, Unreal 2 didn't include multiplayer out-of-the-box. There is, however, a class-based CTF mode available as an add-on.

Unreal 2 Special Edition, however, does include multiplayer out of the box. And the downloadable XMP says it requires the PLAY DISC. The GoG.com version obviously doesn't include this.

Oh ho! In that case, the difficulty may be due to the fact that the official master server died some time ago. This thread on the BeyondUnreal forums has instructions on setting up Unreal2 to use a fan-run server (assuming that one's still alive...)
Edit: Three other master servers listed in the instructions here.

How do I get the XMP in the first place? I still don't have a PLAY DISC which the installer says it requires.
Also, why does this thread say solved? I didn't mark it solved.

OK, I'm working from the Unreal Anthology version, rather than the GOG one, which is why all answers have been shots in the dark...
There's a "Community Edition Installer" which doesn't require a play disc (anyone with the Unreal Anthology version has the same issue). It also sets up at least one of the user-run master servers as well, and includes a few custom maps. I'm not sure the legal status of it (since it will let you install XMP without U2), so I won't link. Consult Google, it's easy to find.
You shouldn't need it at all, unless the GOG version really doesn't include XMP, which wouldn't surprise me considering the lack of master server support. If it IS included, it's an entirely separate program. It doesn't show up as a "multiplayer" option in the main Unreal 2 menu, if that's what you were expecting.
One of the GOG admins marked the thread.

I also noticed that today. Broken link?