PlayStation = Hardcore, Xbox = Casual, Nintendo = Kiddie?
This industry still thrives on labels and stereotypes, whether we like it or not. It's what tends to fuel the most discussion because of course, those who participate in the hobby have immensely strong opinions.
As we look at the current console market today, it's difficult to apply only one word to each manufacturer. Obviously, it's not really fair to do so but in the interest of conversation and argument, it's always interesting to consider.
Over the years, stereotypes bend and change with the times. However, perhaps the labels listed in the title are more enduring than anyone cares to believe...
Nintendo = For the kiddies?
Oh, people get all bent out of shape over this one. But It's clear to anyone with eyes that Nintendo targets a lower age demographic with their products, as opposed to Sony and Microsoft. I mean, that's just logic and common sense. I think what really bugs people is the idea that Nintendo is exclusively for kids, which isn't true. That would be their intended audience for the most part, but it doesn't mean adults can't have fun with those games.
That's why the label is so insulting to Nintendo followers, especially to those who grew up with the iconic brand. It's not because they necessarily disagree that Nintendo products are family-friendly; it's just because some narrow-minded gamers mock these more accessible, inoffensive adventures for being "kiddie." At the same time, there's a reason Nintendo manufactures such software; it remains the ultimate "parent brand," so-to-speak. If you've got young children, chances are, a Nintendo platform will be your best bet.
That's not insulting; it's just reality.
Xbox = For the mainstream gamers?
This is another hot-button issue, mainly because of the preponderance of crossover titles between PlayStation and Xbox. As the vast majority of games produced end up on both platforms, how can anybody argue that one is more "casual" than the other? Well, once again, we look at the exclusive software to make certain determinations. You can also examine the hardware as well; for instance, the Xbox One features a nifty TV-oriented feature that has nothing to do with video games, and isn't anything a lot of core gamers care about.
Xbox is big on multiplayer and always has been. Now, you could say multiplayer entertainment is just as "hardcore" as single-player but I don't agree with that. The roots of multiplayer entertainment are social in nature; they spring from LAN parties and general parties where simple, easy, accessible gaming was the focal point. Yes, I know MMORPGs are a big part of the environment and they're basically the epitome of hardcore. But in the console world, multiplayer often takes on very different meanings and connotations.
Single-player entertainment we always equate with the loners, those who choose to experience their hobby by themselves. And these people are far more likely to be considered "hardcore" than those who play a few rounds of Halo or Call of Duty here and there.
PlayStation = For the hardcore "purists?"
Continuing in the same vein as above, let's look at the software: Recent exclusive PlayStation titles tend to put a premium on the single-player experience, and perhaps that's precisely why this brand draws the so-called "hardcore purists." Heavy Rain, Demon's Souls, LittleBigPlanet (commonly misconstrued to be kiddie or casual due to its cutesy exterior, but extremely in-depth), Uncharted, God of War, even Killzone (one of the few FPS franchises in the previous generation with good campaigns); these are good examples. And let's not forget that Metal Gear Solid and Final Fantasy once were PlayStation exclusives as well.
And look what's coming: Bloodborne, The Order: 1886, Uncharted 4: A Thief's End, The Dark Sorcerer (or whatever Quantic Dream is working on) and hopefully, at some point, The Last Guardian. These are very often considered "core" titles and would not necessarily appeal to the casual mainstream masses. They also typically represent the cutting-edge when it comes to technology in the console world; another trait commonly associated with anything "hardcore."
Now, these are only arguments that support the labels in question, and I do not necessarily agree with everything written. It's just a presentation of what so many believe, and offers everyone a chance to rebut. :)