I don't get into the Fire Emblem stories as much as some fans, but from what I've put together from the games I have played, it all takes place in the same overall history, though generations apart. I can say that with Final Fantasy, officially, every game's story takes place in it's own separate universe with no connection except Dissidia and a few dimension hoppers like Gilgamesh (spinoffs of individual games are the exception, like FFX-2). I would bring up Zelda, but that's inviting trouble and I'm already guilty of derailing the thread a bit.
Back on topic, my thoughts are that general overarching ideas get carried from game to game, but I currently take the view that each game is it's own entity. The whole "it's like Looney Toons or Disney cartoons" thing puts things like Mario Tennis, Mario Party, Mario Kart, and all those other games into a different perspective. Those are just their own little worlds, kinda. While we're at it, one thing about the DKC series is how it breaks the 4th wall all the time. That sort of got into the Mario games too, with things like Donkey Kong being jealous of Mario's popularity and action figures and wanting to steal them all, or Wario starting his own video game company, and that sort of thing.