And they say that chivalry is dead; unfortunately, it's not.
This started out as a post about some of the flaws with The Legend of Zelda series, until I realized that one of the main flaws I was going to bring up actually is a pattern that crops up in a figurative ton of Nintendo titles. I figured I might as well talk about the broader issue in this case, even if it makes me look like a Nintendo hater. (I'm a grumpy gamer, never a hater.)
First, we have to establish just what Nintendo loves: the "lady and knight" character dynamic. (Yes, I'm linking to TVTropes, deal with it.) In this case, the more specific version they use is that the player character is the "knight", while there is always a lady who you work to protect and help - which, in almost every case, involves rescuing them.
Obviously, The Legend of Zelda has been this way from the very beginning. Only a handful of games don't have Link being the protector of Zelda, and those either have him being the "knight" for another "lady" (Twilight Princess, Oracle of Seasons), imply it was that way offscreen (Link's Awakening),and/or have a different "lady and knight" combo (Majora's Mask - Lulu and Mikau, Oracle of Ages - Nayru and Ralph).
The same goes for almost all of the Super Mario Bros. games and their non-sports based spinoffs. Super Princess Peach and the Luigi's Mansion games are the only exceptions I can think of. (There's probably at least one other I'm forgetting, but the exception doesn't disprove the rule here.) If you were expecting some other character dynamic, you'd be better off looking in another castle.
In addition, many games in the Fire Emblem series invoke this. I'm not as familiar with them, having only played a few games in the series. But given the medieval fantasy that most of them take part in, knights are everywhere, so this is the least surprising. I'm a good deal more forgiving of those - or I would be, if "ladies and knights" weren't everywhere else.
Kid Icarus: Uprising is built around the same dynamic, between Pit and Palutena. Even Kirby got his own "lady" of a sort in Kirby 64: The Crystal Shards, in the form of the fairy Ribbon. Advance Wars: Days of Ruin had Will and Isabella. Oh, and don't think I've forgot Pokémon. Sun and Moon make the trainer the "knight" (although admittedly with the slightly better caveat that you can play as a girl) and Lillie is the "lady".
Metroid is the only major franchise to basically have never touched this dynamic - and Nintendo looks like they've given up on Samus's adventures in general for now, so...yeah. Let's just move on.
I've come to suspect that lot of the people who have been clamoring for a female Link don't specifically want such - what they want is the "lady and knight" dynamic to not be repeated yet again. And despite this. Nintendo's handling of E3 2016, along with the story of Breath of the Wild, tells us they're going to keep doing this - in Zelda, in Mario, and likely in other game franchises as well. It's not a set-in-stone trademark, but it is a worrying trend.
Don't get me wrong, Nintendo is capable of making games with great gameplay, and pushing the envelope in many ways in that area. But story-wise, it's starting to look like they are lagging further and further behind - they are doing nothing to advance the craft of video games by recycling the same stories over and over.
Also, this isn't about sexism, although cleaning up that bugbear would be an added bonus. It's about variety - Nintendo offering new stories with new relationships and potentially new characters. Which, if they keep insisting on "lady and knight" based plots, isn't going to happen. Then again, the games over the next couple of years might tell a different tale. Let's see...what about Super Mario Odyssey...
...oh wait, the implied premise is Mario having to rescue Peach from getting married to Bowser. Never mind.