So many options. Some deep cuts for you though:

PlayStation:

Rakugaki Showtime - a bizarre four-player arena battle with a ton of zany unlockable characters, this game was never translated but is worth the effort since it was made by the gods of game design, Treasure (makers of Gunstar Heroes, Contra Hard Corps, Silhouette Mirage, Ikaruga, and many other gems - made up of some of the OG designers of Contra).

Trap Gunner - two player split screen 3/4 view game where you lay traps to try and kill the other player. Not a ton of longevity, but worth several fun afternoons.

Blast Chamber - four player frenzied dystopian sportsball game that takes place in a cube-shaped arena with a clever gimmick: any player can run up to a wall and rotate the entire arena 90° and cause their opponents to fall.

Return Fire - frantic two-player capture the flag with a variety of military vehicles and a fun classical soundtrack.

The Unholy War - made by Toys For Bob, which was at the time the home of the creators of Spyro, Skylanders, Archon, and the greatest PC game of all time, Star Control 2 (now free on Steam as Free Stars: The Ur-Quan Masters). A beautifully balanced and fun combination of chess-like board tactics culminating in asymmetrical 1-on-1 live combat.

Genesis/Mega Drive

Gunstar Heroes - a cooperative run and gun game that out-Contra’s Contra. The game that put Treasure on the map.

General Chaos - weird, small scale real time strategy action with cartoony soldiers. Strange but quite fun.

Star Control - two-player strategy/action in a similar formula to Unholy War (these guys loved this formula and made many variations of it), this game has you navigating a small star map to position your fleet. Conflicts are resolved in a top-down ship combat mode that is incredibly fun on its own thanks to a wild selection of unique ships.

Mutant League Football/Hockey - two versions of the same concept, these games really liven up the sports for those who aren’t die-hard Madden junkies. Most of my games I ended up winning by default because too many of the opposing team members had died. Playing against a friend, you really feel it when you get one of your players murdered.

Super Nintendo

Metal Warriors - split-screen, asymmetric, side-scrolling mech combat. Always fun to get the drop on each other.

Bomberman - obviously not a deep cut, but I had to mention it. Controller-throwing four-player competitive deliciousness.

Uniracers

Chu Chu Rocket on Dreamcast was my favorite multiplayer game that almost nobody I meet has ever even heard of.

Action Half-Life, best mod for goldsrc

I still play the original Wizard of Legend with my brother today. Extremely fun coop rogue like. I can also suggest Rain World though its not nearly as obscure anymore. Rain World is a very unique game.

Wife and I have really been enjoying The Planet Crafter. There is no PVP or fighting enemies, it's all about surviving on a harsh planet, and terraforming it.

Like 20 years ago some pizza chain gave out a very cheap but very well made local multiplayer game similar to Mario Party for Xbox (like, maybe legit just Xbox, the very first one)

Everything was janky, but in a consistently janky way.

So if you picked it up for the first time, it didn't make sense. Especially because it was just a hodgepodge of random minigames with the only consistent thing what color your avatar was. But it didn't take long to learn the game logic, and the games were fun and had replay value. The most memorable thing is they put you in those hamster balls from American Gladiators, and that's what was in the disc label.

I'd be really surprised if there was a ROM of it today, or if anyone even remembers it. But for like the year they were giving them out, everyone had it.

For 5th gen, check out Loaded and Re-Loaded on the Playstation. Diablo-ish.

https://en.m.wikipedia.org/wiki/Loaded_(video_game)

For 4th gen, Gain Ground on the Sega Genesis.

https://en.m.wikipedia.org/wiki/Gain_Ground

One of the last of the many great shooters on the PC Engine is the one-or-two-player Ginga Fukei Densetsu Sapphire.

Ballerburg

Team Buddies is a ton of fun. Here's a clip of the vs gameplay