It's hard to describe but you can usually find it in almost all Smash Bro, Mario Kart, and Mario platformer games.

It's like the magic you get from watching Star Wars, braveheart, saving private Ryan, or any great form of art work that lives up to the high standards of its medium...most people feel the nintendo "Quality" or "Magic" in just how good the gameplay feels interconnected with game design...most nintendo games hit the right notes and score 90s or above with professional game reviewers and the majority of unbiased consumers

its difficult to find a developer whos consistently done that since the 1980s till now, hence the "magic" continues

The game actually working on launch day

But is this a phenomenon you would only attest to Nintendo games? What's with other games of similar quality?
I never played the early Sonic games in my youth, but when I play them nowadays I get so much joy from it. I can hum the music like I knew it since 20 years and smile about the various character animations or feel the guilt when i can't reach an air bubble in time.

I know that Nintendo is a profit-oriented company like all others, but I still feel like they put more heart into their games than some other publishers/developers just milking their franchises for profit, especially nowadays. This might be due to Nintendo's history. They once completely ruled the video game market and did not have to worry about their future if the Virtual Boy or the 64DD was a financial failure. They could do what they want and try out what might be fun or not.

SEGA did the same but their customers lost faith in them at some point.

That said, Nintendo still uses lots of nostalgia by keeping their old franchises alive. The new Zelda or Mario game would not sell as well if it would star a new character as the hero.
Smash Bros. is nostalgia to the max. They put in characters from their whole history in there (Mr. Game & Watch, R.O.B., Villager), add stages and music remixed from old classics and people love it. There's a character for any Nintendo fan in there.
If you've been isolated from Nintendo stuff all your life, chances are you would not care about Smash Bros. at all.

I'd rather call it gaming magic, if games just manage to hit the right button inside you. Like movie magic or stuff like that.
Nintendo is just pretty good doing that over and over again.

Doesn't mean that a company, no matter Nintendo, Square, Bioware, Bethesda or Bungie does that with you personally though. Tastes are different.
But the first gaming, i had that feeling was a Link to the past and i had been gaming years before that. And Yoshi's Island hit me at a point, where i actually was all about PS1, had already played Doom, Descent etc. on PC and really wasn't that much interested in platform games any more.

This is nothing Nintendo exclusive and while some love Nintendo games others might give a fuck about it. But Nintendo always was really good at touching millions of gamers, over and over again.

I wouldn't call that Nintendo magic but i'd still call Nintendo masters of gaming magic.

It´s imposible to define. And i feel sad for those who can´t feel it.

One phrase : The use of subliminal messages in their games.

I'm joking of course

I agree with some of the above posters that have stated this "magic" isn't limited to Nintendo games.  I loved the early Final Fantasy games (ok the first 9 games) to death, they, like Legend of Zelda and Metroid games are among my most favorite of all time.  Part of that has to do with the fact that growing up in the 80's many of these concepts / gameplay ideas where brand new to me (and some to the rest of the world as well).  That said most of those games are still fun to me today.  Not everything I loved from my childhood still holds up, a lesson I learned when I tried to watch the first season of Knight Rider a couple of years back.  I loved that show as a kid, as an adult I could not make myself sit through it (sorry I don't mean to piss off people that still like the series, I am just relaying my inability to stomach it these days).

In conclusion I'd say it is called Nintendo magic not because they are the only company that can deliver said magic but because they are the company that, thus far, has delivered it the most consistently and often.

Except that there was no Xenoblade Chronicles when I was a kid, yet it still holds the Nintendo magic for me, so nostalgia can't explain it.

It absolutely is a phenomenon that only applies to Nintendo, but that doesn't mean that Nintendi is the only publisher to make great or impactful games. Bioshock is one of my favorite games of all time, and I feel something incomparable when ever I play it, and I like it more than many Nintendo games, but it does not have Nintendo's magic. It just doesn't. Similarly, Kill Bill is one of my favorite movies, but it could never have the magic that Disney's The Little Mermaid has, even though I like the former more than the latter.

Saying that Nintendo has magic isn't saying that other games are less compared to them, but there's a special quality to a Nintendo game. It's a very specific thing and it's not explainable. There's a soul to Nintendo games that other games just don't have.

It has nothing to do with nostalgia. You can play a Nintendo game, not knowing it's a Nintendo game, and know for sure after 10 minutes that it's a Nintendo game. You can feel that. It's not about the characters. I felt that way with Dillon's Rolling Western. Many people feel that way about Pikmin. You can already see glimpses of it in Splatoon. Smash Bros is a game built on fanservice. That much is obvious, but it's "magic" doesn't at all come from having Mario and Link duke it out. If you were never raised on Nintendo, there'd obviously be nothing compelling you to buy the newest one, but I guarantee you that if that person sat down and played it with three other friends for an hour, they'd get it. I've seen it happen, and I'm sure you have to. They'd probably be getting their ass kicked, but they'd love every minute of it.

Some Zelda titles made me feel like this (Link's Awakening, Ocarina of Time, ESPECIALLY Majora's Mask, and Wind Waker), along with pretty much all the Fire Emblem's (especially if you play them in classic mode) go into character detail. Granted Mario games are shallow (expect for Super Mario RPG), but that's why LoZ and FE are my 2 favorite series, because they go into LOTS of character detail like the games you've mentioned.