A really good example of this is Guilty Gear X2's story mode. Certainly, it is a very different experience from Subspace Emissary; GGX2's story mode is simply a progression of one-on-one fights with dialogue taking place between fights to further the plot, as well as a few branching paths in each character's story. However, GGX2's story mode accomplishes something very important: it fleshes out the characters and world way beyond the norm for fighting games. After playing through the story mode, a player develops a better understanding of the characters and becomes more attached to them.
This effect can also be seen in FPS games like Halo. First person shooters are built around multiplayer deathmatches, much like fighting games. However, Halo's story mode is one of its strongest features, even though it involves very different mechanics and structure than the it's multiplayer mode, much like Subspace Emissary does. Without its epic story fleshing out Master Chief's character, the SPARTANs would look like just another kind of space marine. However, that story did create and develop popular characters and built up a strong fan-base for the franchise.
One can argue that this treatment is unnecessary for a game series like Super Smash Bros. which is based on a cross-over of established characters from other series. These characters already have an identity of their own. However, not everyone has played all of the games these characters are from. For example, Lucas and Marth are from games that so far have not been released outside of Japan, and other characters like Pit and Ice Climbers are from very old games. So, it can't be assumed that a player necessarily knows anything about most of the characters. Furthermore, some of these characters (such as the two older ones I mentioned above) were never given much in the way of character depth or personality in their original games. So, there is merit in trying to establish who the various characters in the game are, even if it is just to raise the player's interest in other games.
So, how well does Subspace Emissary do this? Despite having no dialogue whatsoever, I think it does do a pretty good job. The cinematic scenes manage to inject a lot of personality into many of the characters. My favorite example is a scene where Ike, Marth, and Meta Knight observe a monster passing below them as they stand atop a cliff. Ike immediately jumps off the cliff to pursue, leaving Meta Knight and Marth staring at each other in surprise. Meta Knight then turns and follows, while Marth looks rather exasperated for a few moments. This scene establishes that Ike is a gung-ho type of character, while Marth is more of a "let's think things through" type. There are plenty of scenes like this through Subspace Emissary.
However, Subspace Emissary does not do everything perfectly. Most of all, each individual character does not get much screen time. Four characters in particular only appear in brief intro scenes. I would have probably followed in the footsteps of Guilty Gear X2 and given each character their own unique story path. That way, every character gets some time in the sun and a chance to be developed a bit. It would have been a good excuse to throw in some more mode completion character trophies too.
On that note, some of the development of character identity is done by the descriptions of various trophies. However, the discussion of Smash Bros. trophies might be worth a post in of itself.