You're thinking like a PF fan, and not like a 339er. Which is fine, I don't expect people to think like 339ers, but it means you are making assumptions that are very PF. Those assumptions are not the assumptions I work off of.
classes are entirely metagame. people multiclass, and it doesn't actually require they make any change in character. If you want tp tell tour story that way you can say you're using the same magic you have been using, even if for book keeping, your charsheet says both "incarnum" and "psionics." No one else can see your charsheet, and who is to say that your own mahic works any way but the way you say it does?
I was more referring to how in universe, magic comes from different sources and works different ways, so people
will have categories for magic users that all get magic in similar ways. Like if you magic comes from your bloodline, meaning you don't have to pray, weren't cursed, and don't have to study, then you aren't going to be called the same thing as a person who has to study magic extensively to even be able to cast relatively minor spells.
Chullivan D&D worlds don't have guys who go out into the woods and commune with nature to become a Druid. They have guys who go out into the woods and commune with nature to become a "druid," but that's mechanically represented by Totemist2/Ardent10/Soul Manifester 8 or Formbound Surger5/Anyform Savant 10/Phenotype Impresionist5, or etc. Or you can actually be a Druid based build, but that's just a happy coincidence in nomenclature.
If all of those get spells, in universe, the same way, and all cast in similar manners, then there is going to be a name for them for people to understand and lump them together.
I mean, if you houserule that it doesn't matter how a class states they gain spells, you can flavor it any way you want, then yeah, a name based on a class makes less sense. But saying that your character can cast spells without studying a book, even if your character sheet says you have to study a spell book, is kind of breaking the rules and screwing any attempt at power balance. Like, if a wizard 10 character just says "Oh, I wake up and know spells in the morning, cause my god gives it to me," that is apowerful upgrade to class powerlevel.
Your powerset determines if you're Thor or Ironman, but the mechanical method you use to get there is entirely opaque. Both Thor and Iron Man can be using the Ravaged Soul class to provide powers that thematically resonate with and allow them to perform the feats of Thor and Iron Man. Or Thor can be a Dragonfire Adept while Iron Man is a Formbound Spiritist.
If those classes have rules for how their spell casting works, but people just ignore it, then yeah, there's no point in a name since the universe stops making rational sense. (Note, I don't know those classes, and google wasn't giving me any helpful links to anything but the Formbound Spiritist).
Even if flavoring abilities is allowed, I still prefer that things work in similar manners, and thus even if the classes are different, there will still only be a handful of ways to get magic, thus people will still be connected by a name for how they get magic (or claim to get magic).
Nothing says Iron Man has to be an Artificer or even mechanically be wearing armor. You can represent the suit using a suite (lol) of approrpiate powers.
This just seems silly to me. How do you even have balance then? I want to play a wizard, but I want to flavor it like a Cleric, can I just sleep and wake up with any wizard spell I want? That's hilariously overpowered. Why would anyone ever play anything, arcane or psionic, that didn't just get abilities like a Cleric then? Since it's all flavor?
Class agnostic can be hard for people used to playing Mideval Tolkeinesque to wrap their minds around, but well, thay's just how I roll.
I mean, even in DnD 2e, you still had to be a specific type of character to level up. Like a Druid couldn't take more than 14 levels without becoming the leader of all druids, IIRC. And Fighters after a certain level became literal lords of estates, and their class name in universe changed to reflect that.
Also, I'm not saying that if you are a Wizard, you will call yourself a "Wizard," I'm saying that in universe, if a bunch of people have to read a book to get spells, they will be called something. Likewise with people that seem to have magic voice powers that come out through performances. Etc.