At least with Linux and Windows ports.

Free:
0 A.D.
play0ad.com/
Alien Arena
red.planetarena.org/
Assault Cube
assault.cubers.net/
Assault Cube Reloaded
acr.victorz.ca/
Board Game Arena (not all but many games are free)
boardgamearena.com/
Bob's Game
bobsgame.itch.io/bobs-game-puzzle-game
Cube 2: Sauerbraten
sauerbraten.org/
Counter-Strike 2D
www.cs2d.com/
Flare
flarerpg.org/
FreeCiv
freeciv.org/
Hase
ziz.openhandhelds.org/hase/index.htm
Hedgewars
www.hedgewars.org/
Invisigun Heroes (free guest edition with three characters and a bunch of maps)
sombr.itch.io/invisigun-heroes
Liero
www.liero.be/
Minetest
www.minetest.net/
Open Arena
www.openarena.ws
OpenTTD
www.openttd.org/en/
Scorched3D (current Linux version may be hard to install, Windows version also perfectly works via WINE though)
scorched3d.co.uk/
Shadow Warrior Classic Complete
www.gog.com/game/shadow_warrior_complete
Teeworlds
www.teeworlds.com/
The Battle for Wesnoth
www.wesnoth.org/
Thread Marks
www.longbowgames.com/treadmarks
Triple A
www.triplea-game.org/
Urban Terror
www.urbanterror.info/home/
Warsow
www.moddb.com/games/warsow/downloads
Warzone 2100
wz2100.net/
World of Padman
worldofpadman.net/en/
Xonotic
www.xonotic.org/

Commercial:
AI War: Fleet Command
www.gog.com/game/ai_war_fleet_command
Airline Tycoon Deluxe
www.gog.com/game/airline_tycoon_deluxe
Anomaly 2
www.gog.com/game/anomaly_2
Apotheon
www.gog.com/game/apotheon
Aragami
www.gog.com/game/aragami
Ascendant
www.gog.com/game/ascendant
Baldur's Gate: Enhanced Edition
www.gog.com/game/baldurs_gate_enhanced_edition
Baldur's Gate II: Enhanced Edition
www.gog.com/game/baldurs_gate_2_enhanced_edition
Barony - Cursed Edition
www.gog.com/game/barony_cursed_edition
Battleworlds: Kronos
www.gog.com/game/battle_worlds_kronos
Broforce
www.gog.com/game/broforce
Capsized
www.gog.com/game/capsized
Chaos Reborn
www.gog.com/game/chaos_reborn
Conga Master
www.gog.com/game/conga_master
Daikatana
https://www.gog.com/game/daikatana
https://bitbucket.org/daikatana13/daikatana
Death Road To Canada
www.gog.com/game/death_road_to_canada
DEFCON
www.gog.com/game/defcon
Descent
www.gog.com/game/descent
www.dxx-rebirth.com/
Descent 2
www.gog.com/game/descent_2
www.dxx-rebirth.com/
Din's Curse
www.gog.com/game/dins_curse
www.gog.com/game/dins_curse_demon_war
Divinity: Original Sin - Enhanced Edition
www.gog.com/game/divinity_original_sin_enhanced_edition
www.gog.com/game/divinity_original_sin_enhanced_edition_collectors_edition
DOOM II + Final DOOM
www.gog.com/game/doom_ii_final_doom
www.zdoom.org/downloads
zdoom.org/wiki/ZDL (for Co-Op with GZDoom)
zandronum.com (also ideal for playing the campaigns in Co-Op)
odamex.net/
DROX Operative
www.gog.com/game/drox_operative
www.gog.com/game/drox_operative_invasion_of_the_ancients
Dungeons 2
www.gog.com/game/dungeons_2
www.gog.com/game/dungeons_2_a_chance_of_dragons
www.gog.com/game/dungeons_2_a_song_of_sand_and_fire
www.gog.com/game/dungeons_2_a_game_of_winter
Dustforce DX
www.gog.com/game/dustforce_dx
FlatOut
www.gog.com/game/flatout
fo-online.net/
FlatOut 2
www.gog.com/game/flatout_2
Fragile Alliance
www.gog.com/game/fragile_allegiance
Frozen Cortex
www.gog.com/game/frozen_cortex
Frozen Synapse
www.gog.com/game/frozen_synapse
www.gog.com/game/frozen_synapse_red
www.gog.com/game/frozen_cortex_mega_tier
www.gog.com/game/frozen_cortex_ultimate_tier
Gratuitous Space Battles 2
www.gog.com/game/gratuitous_space_battles_2
Hammerwatch
www.gog.com/game/hammerwatch
www.humblebundle.com/store/hammerwatch
Invisigun Heroes
sombr.itch.io/invisigun-heroes
King Arthur's Gold
www.gog.com/game/king_arthurs_gold
Megamagic: Wizards of the Neon Age
www.gog.com/game/megamagic_wizards_of_the_neon_age
Moon Hunters
www.gog.com/game/moon_hunters
Mount & Blade: Warband
www.gog.com/game/mount_blade_warband
www.gog.com/game/mount_blade_warband_napoleonic_wars
www.gog.com/game/mount_blade_warband_viking_conquest
Multivinia
www.gog.com/game/multiwinia
Oh...Sir! The Insult Simulator
www.gog.com/game/ohsir_the_insult_simulator
OlliOlli2
www.gog.com/game/olliolli2_welcome_to_olliwood
Quake: The Offering
https://www.gog.com/game/quake_the_offering
http://nquake.com/
Quake II: Quad Damage
https://www.gog.com/game/quake_ii_quad_damage
http://skuller.net/q2pro/
Quake III: Gold
www.gog.com/game/quake_iii_gold
ioquake3.org/
Race The Sun
www.gog.com/game/race_the_sun
Risk of Rain
www.gog.com/game/risk_of_rain
Satellite Reign
www.gog.com/game/satellite_reign
Screencheat
www.gog.com/game/screencheat
www.humblebundle.com/store/search/search/screencheat
Sir, You are Being Hunted
www.gog.com/game/sir_you_are_being_hunted
Skulls of the Shogun
www.gog.com/game/skulls_of_the_shogun
Slipstream 5000
www.gog.com/game/slipstream_5000
Starbound
www.gog.com/game/starbound
Star Ruler 2
www.gog.com/game/star_ruler_2
www.gog.com/game/star_ruler_2_wake_of_the_heralds
Terminal Velocity
www.gog.com/game/terminal_velocity
Terraria
www.gog.com/game/terraria
The Adventures of Shuggy
www.gog.com/game/the_adventures_of_shuggy
The Original Strife - Veteran Edition
www.gog.com/game/strife_veteran_edition
Trine - Enchanted Edition
www.gog.com/game/trine_enchanted_edition
www.humblebundle.com/store/trine-enchanted-edition
Trine 2
www.gog.com/game/trine_2_complete_story
www.humblebundle.com/store/trine-2-complete-story
Trine 3
www.gog.com/game/trine_3_the_artifacts_of_power
www.humblebundle.com/store/trine-3-the-artifacts-of-power
Torchlight II
www.gog.com/game/torchlight_ii
Two Worlds - Epic Edition
www.gog.com/game/two_worlds
UnEpic
www.gog.com/game/unepic
www.humblebundle.com/store/unepic
Unreal Tournament 2004 Editor's Choice Edition
www.gog.com/game/unreal_tournament_2004_ece
pcgamingwiki.com/wiki/Unreal_Tournament_2004#Install_Native_Linux_version
www.gog.com/forum/unreal_series/ut_2004_run_natively_on_64_bit_ubuntu
Unreal Tournament GOTY
www.gog.com/game/unreal_tournament_goty
pcgamingwiki.com/wiki/Unreal_Tournament#Availability
Ultimate General: Gettysburg
www.gog.com/game/ultimate_general_gettysburg
Victor Vran
www.gog.com/game/victor_vran
War for the Overworld
www.gog.com/game/war_for_the_overworld_anniversary_collection
www.gog.com/game/war_for_the_overworld
Whacky Wheels (local only)
www.gog.com/game/wacky_wheels
Windward
www.gog.com/game/windward
Zombasite
www.gog.com/game/zombasite
www.gog.com/game/zombasite_demo

Zod Engine doesn't count, right?

Multi-platform via WINE (tested):

Heroes of Might and Magic 3 - Complete Edition
www.gog.com/game/heroes_of_might_and_magic_3_complete_edition
Mega Man 8-bit Deathmatch (freeware)
cutstuff.net/mm8bdm/
Soldat (freeware)
soldat.pl/en/
Super Mario War (freeware, local multi-player only)
smw.supersanctuary.net/site/
www.smwfs.webs.com/ (working download-link)
Worms Armageddon
www.gog.com/game/worms_armageddon
Red Faction
www.gog.com/game/red_faction
purefaction.org/
Red Faction II
www.gog.com/game/red_faction_2
Titan Quest Anniversary Edition (LAN only)
www.gog.com/game/titan_quest_anniversary_edition

As Linux runs on the exact same hardware as Windows and/or Mac or anything else on a computer. I'd be more intereseted in a list of "multi platforms games" with "Computer" as one of the caegories, and some other device as the others. e.g. Mobile phone/tablet, Console X, Console Y, etc. There isn't many, especially DRM free.

I think it would only apply to a handful of Indy games, and they likely don't have cross-platform multi player on a console, but they may on mobile.

One example of a game I played is Star realms which is cross-platform multiplayer. It's on both PC and Android mobile devices. (Maybe Mac/iOS too, I'm not sure.) I don't think it has any DRM, but I haven't checked. You need to log in to use online anyway, as with most games.

Well, because it clearly says, "By playing this game you understand that it is still in an Alpha / Beta stage." during the installation.
Still, it's quite stable. At least I didn't have any issues.

I think that'd be a very short list.
Also: Everything that needs the Google PlayStore is DRMed for me, which is why I only use F-Droid and DRM-free .apk files on my Android device.

You used to be ble to download the apk directly from their website, I'm not sure if you still can.

Replaced the link for Warsow with a working one.
http://www.moddb.com/games/warsow/downloads

I suppose OpenRCT2 (needs original game files) could fit on that list. Multiplayer for RCT still baffles me, but whatever floats people's boats. ;) So that's what that was called. I've been racking my brain because I was trying to remember the name of the game recently. For some reason I though it was called Urban Assault, and that obviously didn't get me anywhere.

Installed it earlier, and had a few good rounds of CTF. People are still playing. That was my go-to FPS multiplayer game along with Red Faction on my old and long defunct G4 Powerbook. :)

AI War: Fleet Command uses the Unity Engine, so it has a Linux "native" build (which is really just the standard Linux Unity dispatcher using Mono to wrap the Microsoft .NET CLR, but the game is self-contained and runs without Wine, so I think this qualifies as native for your purposes).

Descent 1+Descent 2 (no longer sold on GOG (or any other publisher that I know of), but still available via eBay and to customers who previously purchased from GOG) can do both cooperative and deathmatch games. The games are from the DOS era, but their assets can be used in the source ports D2X-XL and D2X-Rebirth to get native Windows/Linux/Mac builds.

For titles available through GOG, wouldn't it be simpler to start by using GOG Search: System=Linux; Features=Coop,Multiplayer? Admittedly, there may be some good games that meet your criteria not sold on GOG, so that list is not definitive, but it's a good start, and it lists titles currently not on your list.

If you want to include games that are only released for Windows, but which Wine runs acceptably, your list becomes much bigger. You can use other sources (such as Wine's AppDB) to determine the Wine compatibility. You would need to use the GOG Search (or equivalent) to exclude Single Player-only games, but since Wine compatibility is probably harder to verify than game features, this seems like a mild limitation.

Added AI War: Fleet Command, Descent and Descent 2 with the respective engine links.
Apparently, GOG offers a Linux package for AI War now.

I chose DXX-Rebirth over D2X-XL because the latter doesn't even show any dates of last updates while the former seems to be more current and active.

Tested and added
Titan Quest Anniversary Edition
https://www.gog.com/game/titan_quest_anniversary_edition

Installs and runs perfectly out of the box on Linux with WINE 2.22 .

Added
Board Game Arena (not all but many games are free)
DOOM II + Final DOOM (via Zandronum or Odamex)
Invisigun Heroes (native)
Super Mario War (free, local only)
Quake III Arena (ioQuake3)
Unreal Tournament GOTY (+Loki port)
Unreal Tournament 2004 Editor's Choice Edition (+Loki port)

Added to the WINE section:
Red Faction
https://www.gog.com/game/red_faction
https://purefaction.org/
Red Faction II
https://www.gog.com/game/red_faction_2

Outsourced section to post 3.

PS: For some reason I can't edit the first post anymore (not even make it shorter), any ideas?

Counter-Strike 2D (free)
http://www.cs2d.com/