One's debut game has nothing to do with how they're represented in Smash. Daisy is nothing but spin-off based. It's the version of her people care about because it's what made her stand out. It's only natural the games that made her worthwhile as a character does that.
Black Mage is still a character in the game as well. That's what characters are, any kind of individual(that includes any class) you see within a work. Every single enemy, class, etc. are absolutely a character. It doesn't make them inherently fleshed out. Contrary to popular belief, a character does not need characterization to be any less of a character(what you're thinking of is simply improving a character, but it's not really the definition at all. By the fact they actually have a proper name, even as a class, they're already defined as a character. There's tons of NPC's without any sort of name but just a design that have less than your typical class, which is still something). It's even common for a random enemy to have more personality than the actual playable character, who is often an individual too in these cases.
Final Fantasy 1's thing is that you could play multiples of the same character to make the gameplay interesting. They didn't stop being a character just cause they're a "class". Also, we have classes in Smash(Pokemon Trainer, Villager, Wii Fit Trainer). Black Mage is a very popular character idea, and not just cause of Vivi. It's a badass spell wielding magician that sounds really amazing on paper, and is fun to work with in their debut game. When you take a look at one of the coolest fanworks around, 8-Bit Theater, you realize that being a Class or having a proper name isn't relevant to making an interesting character. Writing is. And tons of games do not make their character interesting by default.