Master of Magic

Edit: oops. I think that isn't actually multiplayer. I just got mixed up because I thought I saw some mention of someone making an unfinished multiplayer mod for it.

Yeah. Same. But he's on a laptop so, I figure a limited capability. I should get the details on it. He's not a big gamer so I doubt the laptop is a gaming one.

Another shoutout for oldschool Unreal Tournament.
Possibly (and sadly) the zenith of multiplayer arena shooters.

I have to agree to the previous UT fans. UT GOTY (from ’99) and UT2004 are on the top of my favorite multiplayer games list. UT’s moddability via mutators etc, and mods like Infiltration gave me countless hours of fantastic multiplayer action.

I also have fond memories of Quake multiplayer sessions in 1996 via 10BASE2 LAN at home.

My laptop is not a gaming one, nothing special and a few years old, too, but it can run Divinity: Original Sin EE in multiplayer on lower settings (though it does get a bit hot in the process).

How about Trine 2 Complete Story (2013)? Would that still count as old enough?

I recommend both Total Annihilation games, I love both pretty equally. There's nothing quite like playing TA with friends, especially over LAN. Some criticized TA: Kingdoms for whatever reason but its gameplay is just as solid as the original and it has some of the prettiest aesthetics I've seen in an old strategy game.

Also, it's not on GOG (though it should be as it's effectively DRM-free), I recommend checking out Wolfenstein: Enemy Territory. It's a great WW2-themed FPS based off of Return to Castle Wolfenstein's gameplay. It has a metric crapton of player-made maps, it has a surprisingly active community for a game of its age and it's 100% free.

Beleave it or not but Doom has an active multiplayer community, two to be exact, one dedicated to the ZDaemon Sourceport and another to Zandronum, each with their differences that make them shine.

I do seem to recall the enhanced versions of the Infinity Engine D&D games all feature multiplayer

Icewind Dale in particular seems like it would be fun.