Objectively, the Wii by a mile. But for me personally, the N64.

Biggest reason I give the N64 the edge is because all the co-op gaming I did on the N64 were with other kids that also played games as a hobby. The Wii on the other hand, I very often ended up playing with someone who was not a gamer. So a lot of the time I would be stuck having to explain controls or suffer in horror as I'm watching my cousin, uncle, sister, classmate, etc playing the game in the worst possible way lol.

But hard to blame them considering how much the Wii was focused on the casual market.

N64 on the other hand, it was always the gamers gathering at school or with your cousins planning sleepovers and just gaming all night! Goldeneye, Mario Kart 64, Diddy Kong Racing, Smash Bros, Donkey Kong 64, Star Fox 64, any Mario Party, NFL Blitz, Pokemon Stadium, Turok, Perfect Dark, WWF No Mercy, or F-Zero X, all kings at local multiplayer.

No one ever had ALL these games in their collection, so we would find out who had what game and then plan out when we would all go over to play. A lot of these experiences didn't transpire as well on the GameCube or just not at all.

Nintendo 64.
1) Goldeneye.
2) Perfect Dark.
3) Mario Party.
4) Mario Kart.
5) Lylat Wars/Star Fox.

It was a local multiplayer powerhouse back in the day and probably one of the best selling points of the system over the PS1.

Since I haven't really bothered with local Multiplayer to the same extent, with the exception of the original Xbox and Halo 1 and 2.

Now the software libraries and controllers vary. But a massive point that makes a Nintendo system around the top is the number of players supported. And in that regard, there are 4 winners.
DS: Up to 8 players for Mario Kart DS and Bomberman as well as fun multiplayer in general.
3DS: Similar to DS.
Wii U: 5-Player Games (Game Pad and 4 other players), and 8-Player Smash. You can also play virtually every Wii game which while they limit it to 4 players is still a fantastic experience.
Switch: 8-Player Games and an insanely huge library.
Switch might have to win because of the huge library and how much power (by 2017 standards) it has for a handheld.

Nintendo 64 for me.

Wii. N9 other console in history was as good as the Wii for this as it allowed gamers to play with non gamers. Something about the Switch doesn't have that same sort of vibe of play with anyone in split screen.

Home Console = Wii, this was the epitome of local multiplayer with several franchises blowing up during this generation based on their local multiplayer. The Wii's multiplayer was on a whole different level than any other home console ever made. When that console was on that was the happening place. The Wii put the party in party gaming.

Handheld = 3DS, Mario Kart 7 was a gigantic improvement over the DS game, and it made 8 player local multiplayer a standard. With the DS, you could go up to 8 players, but it. would only last about 5-10 minutes before someone got disconnected. Monster Hunter was also a great franchise on this system - IMO, the local multiplayer is what made this franchise great.

Hybrid = Switch didn't have the same volume of highly successful multiplayer games that Wii did, but with its ability to play 8 player local, and 12 player with WiFi available. Much like the 3DS (probably better) it was stable. Mario Kart 8 was perhaps the biggest local multiplayer game of all time.