Sources for those lists:
https://www.gog.com/games?feature=multi&sort=title&page=1
https://pcgamingwiki.com/wiki/
and the respective subforums around here.

Discord for DRM-free multi-player:
https://discord.gg/fs2vyaq

There's actually a terribly outdated old GOGmix about it:
https://www.gog.com/mix/gogs_that_require_galaxy_client_for_multiplayer
Looking through the boards it seems as if prkprkprk has actually left GOG two years ago because of the rise of Galaxy and what obviously came with it.

So lets keep this list up-to-date since it is obviously growing.
Games that need Galaxy for online multi-player and check for unique store accounts before allowing MP:

8-bit Armies + Guardians Campaign
8-bit Hordes
Absolver
Aliens versus Predator Classic 2000
Aragami
Ashes of the Singularity: Escalation + all DLCs
Baseball Stars 2
BattleZone 98 Redux + The Red Odyssey
Blood: Fresh Supply
Craft The World + DLCs
Deathtrap
Descent Underground
Double Dragon Trilogy
Dragon's Dogma: Dark Arisen
Dying Light
Garou: Mark of the Wolves
Gauntlet™ Slayer Edition
Gauntlet™ Slayer Edition - Lilith the Necromancer Pack
Gremlins Inc.
GWENT
Homeworld® Remastered Collection
Homeworld: Deserts of Kharak
Master of Orion 2016
Metal Slug
Metal Slug 2
Metal Slug X
Metal Slug 3
Monster Train
No Man's Sky
Offworld Trading Company
Rise of the Triad (2013)
Roboquest
Serious Sam's Bogus Detour
Shadow Warrior 2
Shock Troopers
Shock Troopers: 2nd. Squad
STAR WARS™ Battlefront™ II (Classic, 2005) - Also has a Galaxy-free multi-player version (classic installer)
Streets of Rage 4
Stronghold Crusader 2
Stronghold: Warlords
THE KING OF FIGHTERS XIII GALAXY EDITION
The King of Fighters '98 Ultimate Match Final Edition (unverified)
The Last Blade
Tooth and Tail
Trine 3: The Artifacts of Power
Twinkle Star Sprites
Vikings-Wolves of Midgard
Witcher Adventure Game, The
Worms W.M.D
---
Games that do not need Galaxy for currently working online multi-player:

AI War: Fleet Command
Age of Wonders III + Eternal Lords + Golden Realms (DRM: needs serial registration and regular authenticity check)
American Conquest + Fight Back (DRM: needs serial registration)
Baldur's Gate
Barony: Cursed Edition
Battle Isle 2
Battle Isle 3
Battle Isle Phase Four: Incubation
Battle Worlds: Kronos (DRM: needs serial registration)
Blood & Gold: Caribbean!
Broforce
Capitalism Plus
Capitalism 2
Celtic Kings: Rage of War
Chaos Reborn (DRM: needs serial registration)
Conflict: Desert Storm
Conquest of the New World
Conquest: Frontier Wars
Daikatana
Darkest Hour: A Hearts of Iron Game
Darkstone
DEFCON (DRM: needs serial registration)
Delta Force
Descent
Descent 2
Descent: Freespace Battle Pack
Diablo Classic (Battle.net)
Die By The Sword + Limb From Limb
Din's Curse + Demon War (DRM: needs serial registration)
Disciples II Gold
Divinity: Original Sin - Enhanced Edition
Divinity: Original Sin 2
DOOM II + Final DOOM
Freespace 2
Grim Dawn
Hammerwatch
Icewind Dale
Incredible Adventures of Van Helsing 1
Jazz Jackrabbit 2 Collection
Neverwinter Nights 2 Complete
Oh...Sir!! The Insult Simulator
Omerta: City of Gangsters + DLCs
Project: Zomboid
Quake 3 Arena
Screencheat
Serious Sam: The First Encounter
Serious Sam: The Second Encounter
Sins of a Solar Empire: Rebellion
Star Wars: Jedi Knight II
Star Wars: Jedi Academy
Starbound
STAR WARS™ Battlefront™ II (Classic, 2005) (classic installer)
Stardew Valley
Terraria
Theme Hospital
Titan Quest: Anniversary Edition
Unepic
Unreal Tournament
Unreal Tournament 2004
Ultimate DOOM, The
Victor Vran
War for the Overworld
WarCraft II: Battle.net Edition
Windward
Worms Armageddon
---
Games that can be played in multi player mode on a single PC via hotseat, splitscreen etc...

Adventures of Shuggy, The
Ascendant
Aqua Kitty: Milk Mine Defender
Battle Chess Special Edition
Battle Isle
Battle Isle 2
Battle Isle 3
Battle Isle Phase Four: Incubation
BlazBlue: Calamity Trigger
Blood: Fresh Supply
Broforce
Capsized
Cat Quest 2
Chaos Overlords
Chaos Reborn
Crawl
Crimzon Clover
Combat Chess
Conquest of the New World
Cultures: Northland + 8th Wonder of the World (open ports 2302 and 1037)
Cuphead (2P)
De Blob
Death Road to Canada
Disciples II Gold
Divinity: Original Sin - Enhanced Edition
Double Dragon Trilogy
Eador: Genesis
Eador: Master of the Broken World
Epic Pinball: The Complete Collection (up to 4P)
Fight'N Rage
Guacamelee
Heroes of Might and Magic
Heroes of Might and Magic 2: Gold
Heroes of Might and Magic 3: Complete
Heroes of Might and Magic 4: Complete
Heroes of Might and Magic 5: Bundle
Jazz Jackrabbit 2 Collection
Metal Slug (2P)
Metal Slug 2 (2P)
Metal Slug 3 (2P)
Metal Slug X (2P)
Mortal Kombat
Mortal Kombat 2
Mortal Kombat 3
Mortal Kombat 4
Never Alone: Arctic Collection
Oh...Sir!! The Insult Simulator
Pinball Gold Pack (up to 8P)
PixelJunk Shooter
Rayman Origins (up to 4P)
Redout: Solar Challenge Edition
Serious Sam: The First Encounter
Serious Sam: The Second Encounter
Shovel Knight
Street fighter Alpha 2
THE KING OF FIGHTERS XIII GALAXY EDITION
Towerfall Ascension + The Dark World
Trine Enchanted Edition
Trine 2: Complete Story
Trine 3: The Artifacts of Power
Tyrian 2000
Victor Vran
Vikings-Wolves of Midgard
Warhammer Epic: Final Liberation
Worms 2
Worms Armageddon
Worms Forts
Worms World Party

Please correct me wherever I'm wrong and mention (preferably tested) games to add to each list.