They are great characters. But the thing about them is just about everything you need to know is right there on the surface. You can look at them all in a crowd, not knowing anything about them, and get a good idea of what their roles are.
I'm sure this image isn't anything new, but I see it every day and it always makes me happy... I only wish it had been made a little later and had Rosalina on it as well.
It's not story that the characters need. It's outlets to express their personalities. Luigi's Mansion gets it, I'd say particularly in Dark Moon. You've got your support guy, E. Gadd, oozing his personality through the little exposition that the game does need. Luigi himself doesn't speak more than a few words, but shows his character in all his animations. How he moves, takes damage, idles, reacts to peeking into windows... it's perfect. Luigi doesn't have to say what he's thinking or feeling, because there it is. And you're the player, you're in the same situation as he is. It doesn't need to be cinematic, or especially detached from gameplay.
To give an example of how the games could do this more where they don't already, I'll use Super Mario 3D World. Anytime control is taken away from the player could have been dedicated to character expression. The way they fall off the screen when they die, or what facial expression and pose they'll have when about to charge into a castle level. The level clear animations are another big one where they could have expressed character, and... there are only 5 playable characters, so, there should only be 30 possible combinations of characters... 5 solos, 10 duos, 10 trios, 5 quartets. What if every combination put the characters into a unique victory pose? I'd love to see this if you cleared a level with just two players as Mario and Peach:
or see Mario and Luigi pose together in a way that plays up the "Bros." part of "Super Mario Bros."
It wouldn't have to be elaborate.
Because, again, who they are is relatively simple, and therefore pretty simple to put together.
No additional words or storylines or conflicts would be needed to get more out of the characters in 3D World. They do already have some of what I'm talking about, but it all kind of blends together and doesn't make the most of the differences between them all. It's not something the game desperately needs or is really worth complaining about, it's just that it's true, there is potential for more of what Luigi's Mansion does right with the characters' personalities that 3D World has a lot of wasted real estate for.