RetroGaming

On the SNES, all of the games below have coop:

• Tiny Toons Wild and Wacky Sports - a collection of several silly minigames as some sort of tournament, with minimum scores that need to be reached. No direct violence between players, only cartoon shenanigans like falling down a cliff, snowballing after tripping while skiing, etc. (Maybe show the kid some of the old cartoons as well, if s/he enjoys, I'm sure the game will be more interesting to him/her)
• Super Bomberman - Might count as "violent", since you're bombing weird looking enemies, but it's a formula that rarely gets old and plays great as co-op. Your kid might enjoy 3 and 4 the most, with the variety of mounts to be had (if your rpi can run NeoGeo games, Neo Bomberman is also a great choice)
• Top Gear - Racing game with one of the best soundtracks in the console. Top Gear 3000 goes to space and has plenty of options for upgrading your car.
• Secret of Mana - Might count as "violent", especially as you start off with a sword and kill cute bunny-thing enemies. An action RPG that, once you get the 2nd character, it can be played with the 2nd controller. Drop-in/drop-out, so you can easily join and leave while your kid plays or vice versa. The same applies for Secret of Mana 2 (Seiken Densetsu 3)
• Magical Quest 2 and 3 - Disney games that are best when played cooperatively (the first lacks coop). Starts off easy enough and you get new costumes that give special powers, which help you out in the stages going forward. It's "Disney violence" how you beat most enemies (jumping on their heads, spinning them against one another)
• Kirby Super Star and Dreamland 3 - Both allow for a second player to join in and help. Dreamland 3 has a very unique and cute looking style.
• International Super Star Soccer - If you're into normal football (soccer), this is easily the best of its kind for the 16-bit era
• NBA Jam - Also available on the Mega Drive, dunno which version is better. BOOM SHAKA LAKA!!

Sonic 2! Let the 5yo be tails! He'll never die. He can collect rings and kill enemies. It's the best 2 player platformer ever created

Goofy Troop

Snake Rattle and roll! For the NES.

Alley Cat Remeow Edition is a fan remake of Alley Cat, an arcade game, with up to 4 players, but it's PC (should still run on that hardware, i hope!)

https://www.youtube.com/watch?v=WIm6oRhyDrM

Lode Runner (NES) might be okay. Not multiplayer, but it's got nice low-stakes puzzling gameplay.

Edutainment games are a bit few and far between on consoles - Donkey Kong Jr. Math springs to mind, but it's remarkably dry for a Nintendo game.

Pacman! I am not entirely sure, but I think it was an Atari 2600 where I really enjoyed this on at a similar age

Troddlers - SNES

The music in Troddlers slaps.

Its a puzzle platformer game similar to Lemmings. Little dudes walk in a straight line and you play as a wizard that can place blocks. You have to get the little dudes to the exit.

The only violence is when the little guys or the other player gets smooshed by a block I think, its been a long time since I played the game.

R.C. Pro Am

Lemmings for Amiga had a 2player split screen mode. Goal is to guide the most lemmings (yours and the other players) into your exit.

SNES:

• Harvest Moon - this can lead into allowing the kid to plant something IRL and having them water it regularly, allowing them to "be a real farmer". Incidentally this is also a great way to get a child to eat vegetables, as a child who refuses to even consider eating a vegetable will change their mind when they grew it themselves

• Super Mario Kart

Mega Drive:

• Dr Robotnik's Mean Bean Machine - simple puzzle game with humorous characters from the old cartoon

Pong

The NES sesame Street games are pretty good.

My youngest loved sesame Street countdown. it was really forgiving and taught her basic platformer mechanics.

It doesn't count as non-violent, but every kid I have played it with has thoroughly enjoyed the TMNT arcade game (as well as the Simpsons game that plays similarly. If you play with them you can essentially carry them through the game and they are just along for the ride, but feel like they are part of the team.

A bunch of arcade games become much more friendly to kids with infinite continues, shmups, fighters, brawlers, etc but for those you'll need to filter on violence. I'd probably also avoid anything RPG or story heavy.

My picks would be Bubble bobble, Kirby on SNES, ice hockey on NES, Mario 1-3, super Mario kart, Tetris, monkey ball, Dr Mario. For Sega Sonic 1-3 & knuckles, ecco, robotnik's mean bean machine, Alex kidd, outrun

A c64 emulator? Then basically anything, LOCO maybe?

Joust. Easily understood game for little ones, and you can discourage player killing.

Kirby Super Star

That's easy. Daggerfall.

Lots of other great suggestions. But I do think the consensus is multiplayer games with coop.

Konami beat ‘em ups like TMNT, The Simpson, Xmen would be great.

I’m also going to throw in Party games or kart racers, CTR and Mario Kart of Mario Party or Crash Bash would also be great

That said Pokémon might be a nice option too though not sure how you’d play that together.

Or edutainment like Gizmos and Gadgets or Jumpstart

Sonic 2.

Controlling Tails can be good fun if you’re very young and don’t fully know what you’re doing!

Can be hard keeping both characters on screen at times though.

Gta 1.

Super Mario World, Battletoads

Magical Quest 2 & 3 are very good for that. They already know the characters, and the games are beautiful and pretty good gameplay-wise.

You play together and if the child loses all his/her lives, he/she can steal yours. For difficult sections or bosses, you can do it alone.

There are new costumes regularly so the child wants to continue to discover the next costume, and its associated powers.

Magical Quest 2 is easier than 3, so I think it’s better to start with this one. You can either play as Mickey or Minnie.

In Magical Quest 3, you can either play as Mickey or Donald. Donald is meant to be played by the child because his gameplay is a bit different (with his soldier costume which is wooden barrel, preventing him to sink while Mickey has a silver armor which kills him if he goes into water with it). It’s not that difficult to play as Mickey (my nephew did not want to play as Donald and we had no major issues).

Goof Troop, for SNES.

Oh, and sorry for so many responses, but I figure that posting these separately will let the votes give you some idea where to start.

The votes I've seen so far very much reflect the order I would try these games out with a five year old gamer.

Fun-N-Games (Genesis/Megadrive) is mostly single player, but has many mini games accessibile to young gamers.

It's single player, but too good to miss: "Richard Scary's Busy Town" (Genesis) is a fantastic open ended set of mini-games that are fun and accessibible to almost all skill levels of gamer.