KMickle: You need GoG Galaxy.
The same here. Neither 'Pixel Ships' nor Multiplayer work. The error messages are "You must be connected to GoG" and "You must link your GoG account to your Wargaming.net" account. Neither option seeems feasible.
tad10: Pixel Ships are disabled, AFAIK even if you do connect your account.. I get that MP error with GoG quite a lot (especially after an MP crash) and have to restart Galaxy to reconnect. Sometimes after manually killing any thread related to Galaxy.
Anyway.
1. Start GoG Galaxy
2. Make sure Galaxy is linked to your copy of Master of Orion (should show up in Galaxy)
3. Start game from Galaxy, not from desktop.
Thanks for the answer.
Actually, I have never been able to use multiplayer; it has been disabled ever since I saw it first. Master of Orion shows up in Galaxy client, but no special options to link it to whatever. Whether I launch manually or through Galaxy, the Multiplayer option is still inactive. Version control shows '03/02/2016'.