I think that having some basic amount of lore/background for each character is pretty important. I'm gonna bring up a fairly obscure example in the form of Rivals of Aether, a platfighter (I know I know, but I promise I'm going somewhere with this) that isn't just a Smash clone, but a game designed to be competitive.
The actual story mode of the game was largely awful, it was generic battles, the CPUs sucked, and the story was entirely forgettable. Beyond that, it was written with only the first 6/12 characters of the game, meaning half the roster isn't even in it. I think this is a good example of how a full story mode night not matter at all; Rivals is a great game despite the dumpster fire that is the story mode.
However, each of the characters has a brief snippet of lore that really fills them out. Clairen is the game's function of a swordie - a character in platfighters with a large disjointed sword hitbox that generally wants to play around the tip of the sword, at midrange. And Clairen is a blast to play. But she becomes a full character when she has the backstory of being a time traveller from an apocalyptic future. Does it really change anything? No, but it makes Clairen more than just a bunch of tipper hitboxes.
Another good example from that game is Elliana, who is a meter management/trap character who is a floaty/airborne heavy with good combos and good kill power, but with that kill power being locked behind her meter. That sentence was largely nonsensical, probably, but I promise it makes sense if you've played Rivals. Elliana is a snake who wanted to be part of the bird air force, but she couldn't fly, so she and her friend built a mech. However she went kinda mad and made it a murder machine, and now she's here for revenge. There's just something about "crazy snake in a homemade mech" that really adds to the necessary insanity that comes with playing Elliana.
Now, a good example of where the lack of lore fails is in Guadua. Guadua isn't technically a character in Rivals - he is a character made for Rivals Workshop, the custom character mode. However, he was made by the developers (as an example character) so it's worth looking at. Guadua in almost every way feels like a Rivals character- he's got a full kit, a unique mechanic and stage control, and a furry design - but he doesn't have a personality or backstory, and he ends up feeling largely incomplete without it. This was kind of on purpose, after all he is a workshop character, but nonetheless I think it demonstrates how some lore can go a long way.