I frequently see people reiterating that Nintendo consoles are great "if you like Nintendo games." But what is a "Nintendo" game? What about the word "Nintendo" makes it a descriptive term applicable to every game published by the company?

If it is not being used as a descriptive term, and is simply referring to games published by Nintendo, then it's a nonsensical statement -- like saying "I like books published by Random House" or "I like movies produced by 20th Century Fox." In this context, the word "Nintendo" must be in use as an adjective, but for the life of me I can't figure out what that adjective means.

Why doesn't anyone say that Playstation or Xbox are great "if you like Playstation/Xbox games" (or any other qualifier which one could attempt to apply to the platforms' entire libraries)? Is the suggestion that "Nintendo" games are all very similar, while Playstation and Xbox provide a much broader variety of content? That answer doesn't sit right with me, because Nintendo publishes a very broad variety of games. In fact, I am hard-pressed to provide an example of two more drastically different experiences than DKC Tropical Freeze and Bayonetta 2. Furthermore, the content available on PS and XB platforms doesn't seem that broad to me, mainly because the output of major Western third-parties (Activision, EA, and Ubisoft) are practically indistinguishable from my perspective. I feel that these three different companies collectively provide a much narrower range of experiences than Nintendo alone.

Maybe I'm approaching this from the wrong angle by focusing on the biggest companies. Playstation has a lot of indie games filling out its catalogue with a greater variety of experiences. But the Wii U has a lot of indie support, too, much of it shared with PS/XB/PC. So we can't allow that to count.

The simple counter to the DKC/Bayo example would be that Bayonetta is only one game, or that the Wii U doesn't have a lot of games filling in the gap between the two. But then you're saying the Wii U is a great console if you don't play very many games, not if you like a particular 'genre' arbitrarily defined as games sharing the same publisher or appearing on the same system.

What do Mario 3D World, DKC Tropical Freeze, Pikmin 3, Hyrule Warriors, Super Smash Bros, Bayonetta 2, Wii Party, Wii Fit, Wonderful 101, Xenoblade X, Splatoon, and The Legend of Zelda all have in common that makes them fit into the same mold? If Bayonetta 2 is a "Nintendo" game, does that mean Bayonetta 1 was also a "Nintendo" game?

The use of this phrase becomes even more baffling when people claim they "don't like Nintendo games." Do they dislike each individual title published by Nintendo after judging them one by one, or does every Nintendo-published game carry a common gene implanted during the development period which renders them entirely unappealing to gamers of a certain persuasion? Furthermore, why does one have to either "like" Nintendo games (in which case the Wii U is a great console) or "dislike" them (in which case it is a terrible console)? Can someone simply not dislike Nintendo games, and would the Wii U be a great console for such a gamer?

Ummmmmmmmmmmm, this could be said about every dev ever.

But I laid out in the OP evidence that this is not the case. Just 2 years into the Wii U's life, Nintendo has published games in the 2D platformer, 3D platformer, puzzle, racing, real-time strategy, action, action/adventure, party, fighting, and fitness genres, and have a shooter and an RPG coming in the next 6 months.

Nintendo Game means 2 different things.
1. Games exclusive to the platform
2. Games developed by Nintendo or its 1st and 2nd party.

When describing it as a supergenre, (toats made that word up), its mainly 2.
When talking about the library of consoles its 1.

Then it is about the publisher. As I said, I feel this is a nonsensical qualifier, like grouping books or movies by their publisher/producer. No one does this.

I think its just the amount of "Polish" that goes into their games... I dont think there is really another developer/publisher that has outputted the same number of consistently polished games as Nintendo does! Their games work at launch and their online also works at launch... There are some bugs/glitches now and then but most are minor and don't usually take away from the full game experience.

But when I say, Nintendo makes great games, I usually refer to First Party games and Second Party games and not really "Third Party" games that are published by Nintendo

But that means saying "The Wii U is great if you like Nintendo games" is literally saying "The Wii U is great if its library appeals to you," which is simpler and more direct to say, and is absolutely no different from the PS4/XB1 (both of which are also great if and only if their libraries appeal to you).

And again, I ask what is so similar about the vast range of experiences Nintendo offers that they can be presented as a single genre, super- or otherwise.

...I'd probably go with Disney, honestly. Good comparison, but it doesn't explain why someone would say "I don't like Disney movies" being inclusive of all of the Star Wars, Marvel, Pixar, etc films produced by them.

I do get your point though, if you like the brand of games Nintendo gives such as games which focus on gameplay over graphics, unique gameplay experiences (such as the gamepad, the wiimote, ect) over story (with the exception of Xenoblade and more recently Metroid and Zelda), and overall fun over competitive play. This is not to say all games Nintendo creates follow this exact guideline to a T, but they seem to share these common themes.

I can't imagine a more incorrect statement. Professor Layton is like a platformer? Zelda isn't much different from Bayonetta?

Generally party games refer to mini-game collections. Game & Wario, Wii Party, etc. Anything with 4-player local multiplayer could also potentially be considered a party game, not so much as a genre but in terms of an appropriate time to play them.