Nintendo's reputation for making children's games is, I feel, undeserved, and the reason why comes down to what do we mean by a 'mature' game? Most of us will have our own idea of what that means, and mine is different to those already mentioned in this thread.
To me, a 'mature' game has nothing to do with graphics, art-style, or story (although these elements can individually be mature) but more to do with design and gameplay. To briefly explain that point I'll use Call of Dutyfor an example. In Modern Warfare 2, the game begins with a tutorial stage that teaches the player how to shoot, aim, throw grenades, and crouch. it walks you through these basic actions slowly to ensure you understand them. However, in Modern Warfare 3, the game begins with the player climbing out of a jeep in the middle of a Russian attack on New York City with no instruction, expecting the player to just get on with it. To me, that approach is more mature, because it leaves, or expects, the player to figure things out for themselves; the game doesn't go out of its way to teach you things which should be obvious or will come naturally. It doesn't 'hold your hand', in other words.
Of course, there is plenty of that in Call of Duty, but that's besides the point. Actually, no, let's roll with that and compare Call of Duty: Ghosts to Donkey Kong Country Returns. The former explains every action to you as it becomes relevant and guides your character automatically at certain parts of the game. Even in multiplayer you are constantly told to 'Press X to climb', or 'Hold B to open door'. The constant repetition of instructions and the fact that it never goes away is similar to constantly having to tell a child to 'hold that with both hands or you'll spill it', or 'tie your shoe laces or you'll fall over'. On the other hand, the latter just sends you off into the game and expects you to finish it. The first level tells you how to do DK's new blowing technique, but that's about it. The game from then on leaves you to do everything alone, even discover advanced techniques like rolling into pits and then jumping in mid-air. Yes, there is the option to have Super DK (or whatever it's called) finish any level for you, but it's optional and is never forced, or even explained to you, in-game. The level design is based around a rhythm, a flow, and players are encouraged to pay attention, because each failure just teaches you a new lesson.
Ghosts is more mature in terms of art and story, sure, but I don't play games for art or story. Of those two examples, Donkey Kong Country Returns is the game more willing to treat the player as a human being with a functioning brain, one who can figure things out for themselves and loves to overcome challenge by understanding how the game is played. It has mechanics that give a helping hand, yes, absolutely, but they are there only if you want them, instead of constantly in operation. For these reasons, I would say that Donkey Kong is a more mature game than Ghosts.
This generally applies to most Nintendo games. The majority of Nintendo's games give you information once and expect you to rememeber it; they are designed to be challenging but in a way that engages, not frustrates (anyone who's done the very last world on Super Mario 3D World will know precisely how well they manage that). Franchises like Advance Wars, Paper Mario, Fire Emblem, Metroid, F-Zero, and Donkey Kong are all more challenging than they first appear. All of them hide complexity beneath a veneer or approachable simplicity, easing you in and then leaving you to do your own thing because you are human, and you have a working brain. When I play Mario Kart, for example, against my younger cousins I see them losing because they don't drift, because they don't know how to do it. I do and so I win. But more importantly, I get to teach them about it and then watch them experiment with it, learn how to use it, grow to appreciate it, fiddle with kart choice to maximise their stats, and then eventually lose to them. The game doesn't teach you that, you have to learn that on your own (or with the help of your cousin) and that to me is maturity in game design.
A lot of the 'mature' triple AAA big-name titles that get released these days are almost insulting by how they treat the player (like when a grenade kills you in Call of Duty and it pops up saying, 'You were killed by a grenade. Avoid these in future,' with a handy little picture of a grenade just incase you forgot what one looks like). Nintendo doesn't do this very often and is the better for it. It happens from time to time--I personally think it happened too much in Zelda on the DS and Wii (*cough* Fi *cough*)--but not enough to make me feel as though I am being treated as a child.
I suppose that's the best way to sum up my feelings; a mature game is one which doesn't make you feel like a child. Nintendo's games, for all of their gay whimsy and colourful cartoon characters, don't make me feel like that. They make me feel smart and skilled and bold. That is why I still prefer Nintendo games over anything else at the age of 23, and why I will still prefer them in ten years time. Nintendo will never be 'kiddy' to me because they don't treat me like a kid; they treat me like a person who can learn and understand things, and that is a pretty mature way of making video games if you ask me.