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There's actually a terribly outdated old GOGmix about it: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.

Games with local multiplayer via LAN:

1nsane
Airline Tycoon Deluxe
Age of Wonders
Age of Wonders: Shadow Magic
Age of Wonders 2: The Wizard's Throne
Alien Nations
An Elder Scrolls Legend: Battlespire
Anno 1404: Gold Edition
Anno 1701 A.D.
Apache Longbow
Army Men RTS
Baldur's Gate Complete
Baldur's Gate II Complete
Battle Isle 3
Battle Isle Phase Four: Incubation
Battle Realms + Winter of the Wolf
Barony: Cursed Edition
Blitzkrieg Anthology
Blitzkrieg 2 Anthology
Capitalism 2
Carmageddon: Max Damage
Celtic Kings: Rage of War
Chaos Overlords
Codename: Panzers - Phase One
Codename: Panzers - Phase Two
Combat Chess
Conflict: Desert Storm
Conga Master
Conquest of the New World
Conquest: Frontier Wars
Constructor Classic 1997
Cossacks Anthology
Cossacks II Anthology
Cultures: Northland + 8th Wonder of the World
Daikatana
Darkest Hour: A Hearts of Iron Game
Darkstone
DEFCON
Delta Force
Delta Force 2
Delta Force: Land Warrior
Descent: Freespace Battle Pack
Die By The Sword + Limb From Limb
Diablo
Diablo Classic
Din's Curse + Demon War
Disciples II Gold
Divinity: Original Sin - Enhanced Edition
Divinity: Original Sin 2
Dying Light
DOOM II + Final DOOM
Dungeon Keeper Gold
Dungeon Keeper 2
Dungeons & Dragons: Dragonshard
Empire Earth Gold Edition
Empire Earth 2 Gold Edition
Freespace 2
Giants: Citizen Kabuto
Grim Dawn
Hammerwatch
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
Icewind Dale
Icewind Dale 2 Complete
Incredible Adventures of Van Helsing 1
Lords of the Realm: Royal Edition
Neverwinter Nights 2 Complete
Master of Orion 2
Magic Carpet Plus
Mortal Kombat 3
Nox
Populous
Populous 2: Trials of the Olympian Gods
Populous: The Beginning
Powerslave (DOS version)
Project: Zomboid
Quake
Quake 2
Quake 3 Arena
Sacrifice
Screencheat
Serious Sam: The First Encounter
Serious Sam: The Second Encounter
Settlers 2: 10th Anniversary, The
Settlers 3: Ultimate Collection
Settlers 4: Gold Edition, The
Sid Meier's Alpha Centauri™ Planetary Pack
S.T.A.L.K.E.R.: Shadow of Chernobyl
S.T.A.L.K.E.R.: Call of Pripyat
S.T.A.L.K.E.R.: Clear Sky
Star Trek: Away Team
Star Trek: Starfleet Command III
Star Trek: Bridge Commander
Star Trek: Elite Force II
Star Trek: Voyager - Elite Force
Star Wars: Battlefront II
Star Wars Episode 1: Racer
Star Wars: Jedi Knight: Dark Forces II
Star Wars: Jedi Knight II
Star Wars: Jedi Academy
Star Wars: Rebellion
Star Wars: X-Wing vs. TIE Fighter
Star Wars: Empire at War: Gold Pack
Stardew Valley
Stronghold HD
Stronghold Crusader HD
Syndicate Wars
Terraria
Titan Quest: Anniversary Edition
Total Annihilation: Commander Pack
Total Annihilation: Kingdoms + Iron Plague
Two Worlds Epic Edition
Unepic
Unreal Gold
Unreal Tournament
Unreal Tournament 2004
Ultimate DOOM, The
Vikings-Wolves of Midgard
War for the Overworld
WarCraft II: Battle.net Edition
WarCraft II: Enhanced
Wing Commander: Armada
Windward
Worms 2
Worms Armageddon
Worms Forts
Worms World Party
(tbc)

Grim Dawn has a LAN option in the multiplayer settings, though I've never tried it.

Some games that have offline multiplayer with a single computer, no Galaxy or network connection required:
Guacamelee
Crimzon Clover
Shovel Knight (with most recent update)
Tyrian 2000 (IIRC)

Thanks to both.
I've thought about a list with single computer multiplayer and will add it since you already started it.

Does anyone know whether the multiplayer mode of "Anomaly 2" and for "8-bit Invaders" works without Galaxy?
It's not stated on the game card but that doesn't mean much.

Offworld Trading Company requires Galaxy for multiplayer

Thanks, added.

Also added "Ashes of the Singularity: Escalation + all DLCs".
The list is getting longer...

Also:
What did become of the Galaxy plans for Aragami?
Is it still possible to play it online without Galaxy?
http://www.gog.com/forum/aragami/multiplayer/post3

Broforce
Victor Vran

Both offer local co-op on the same computer. Unfortunately for the online multiplayer both rely on servers run by the devs and don't support multplayer via LAN.

Thanks, added.

Also added "BattleZone 98 Redux + The Red Odyssey" to the mandatory Galaxy list
and a few more to the others.

I'm also through page one from here now.
It would be nice if someone else would go through page 2, etc... and check the store pages, the subforums and probably also
https://pcgamingwiki.com/wiki which also shows multiplayer types also split between local and local via LAN.
Maybe we could make that a community effort (in the hopes of at least some remnants of the old community being around at all).
Since I get region blocked, some don't show for me anyway.

Online multiplayer of Omerta works fine without Galaxy.

Oh...Sir!! The Insult Simulator offers both a local hotseat vs mode and an online vs mode without the need for Galaxy. Or at least the Linux version does.

Props for your work, +1

You can earn some rep by telling me about the multi-player modes of Blitzkrieg 3:
https://www.gog.com/forum/blitzkrieg_series/blitzkrieg_3_multiplayer_lan_support_galaxy_dependent_online_multiplayer

Blood & Gold: Caribbean!
https://www.gog.com/forum/blood_gold_caribbean/blood_gold_caribbean_multiplayer_lan_support_galaxy_dependent_online_multiplayer

Conga Master:
https://www.gog.com/forum/conga_master/conga_master_multiplayer_local_lan_support_galaxy_dependent_online_multiplayer

8-bit Invaders:
https://www.gog.com/forum/8bit_invaders/is_multiplayer_galaxy_exclusive

Anomaly 2 (I forgot to mark the question box here, sorry)
https://www.gog.com/forum/anomaly_series/multiplayer_ipbased_lansupport_devs_servers_galaxy_mandatory

Cultures 1+2:
https://www.gog.com/forum/cultures_series/cultures_12_multiplayer_lan_support_galaxy_independent_online_multiplayer

Cultures (3): Northland + 8th Wonder of the World
https://www.gog.com/forum/cultures_series/cultures_3_multiplayer_lan_support_galaxy_independent_online_multiplayer

Deadlock 1+2
https://www.gog.com/forum/deadlock_series/deadlock_12_multiplayer_lan_support_galaxy_independent_online_multiplayer

Delta Force Series
https://www.gog.com/forum/delta_force_series/delta_force_series_multiplayer_lan_support_galaxy_independent_online_multiplayer

The game-cards could really include such information...

Finished going through page 1+2 here:
https://www.gog.com/games?feature=multi&sort=title&page=1

Hopefully the information I found on each game in the subforums and pcgamingwiki was correct.
May have missed some region blocked games as always (no idea).

I won't have time to do page three in the next week due to work and other things, maybe someone has.