Barbarian is simply a class in DnD.
One that uses Rage to enhance their abilities and has some level of affinity for wearing light or no armour.
Nothing about the class as a whole dictates background any more than most other classes (Older editions shoehorned classes like Druid and Monks into certain backgrounds with their alignment restrictions - All Druids had to be Neutral in some way and all Monks had to be Lawful in some way)
With the only nod towards "Savage tribe" culture being the typical depiction of an "Unarmoured" Barbarian running around wearing some animal skin loincloth. But you could equally get some sort of Bruce Banner style "You won't like me when I'm angry" type Barbarian that focuses on Charisma and avoiding conflict, but when pushed will enter a battle rage and tear their foes apart with martial prowess.
As far as class vs subclass goes... DnD has been rather fluid over the editions with what constitutes a full on class and what is reserved for subclasses. Back in Pathfinder, Eldritch Knight is a prestige class (A class that has certain requirements to access and only has 10 levels of progression) while in 5e it's a subclass of Fighter. Same is true for Assassin and Arcane Trickster, in Pathfinder they're prestige classes, in 5e they're a subclasses of Rogue.
You can argue Barbarian being a subclass of Fighter just as much as you can argue Ranger to be a subclass of Fighter - Both Barbarian and Ranger are just Fighters with a little extra flavour.
Though, honestly, separating things into individual classes provides more versatility. Not only because you can then split these classes further into more subclasses, but you can also multiclass similar classes to combine aspects of them (For example, in BG3 if I wanted to have a Thief for double Bonus Actions alongside some spell casting from Arcane Trickster I'm SOL. But if they're separate classes, I can multiclass Thief Rogue and Arcane Trickster).
An example of this is my WotR character who was a Mad Dog Barbarian for bonus attack when fighting together with a companion combined with Bloodrider Bloodrager for bonuses when mounted as well as a Bloodline to bolster Rages and Hunter for boosts to companion stats and shared Tactics feats. All of this providing much more than a simple Barbarian + Beastmaster Ranger multiclass can in BG3 (Even if we did have 20 levels to utilize)
