As I just posted in the 'NPCs with class levels' thread, I prefer to think of the PC classes as being pretty unique - they describe the PCs and not necessarily anyone else. Yes there are plenty of barbarians, but none of them have quite the same abilities as the barbarian the PC is playing. Class is a metagame term, and trying to decide which 'class' everyone in the world falls into can lead to some strange contortions. It can be fun in a goofy, gonzo setting, but in a lot of settings I think it is best avoided, in 5E at least.
And to keep on topic, non barbarian barbarians:
A feral youth, raised by wolves and has taken on many of their traits (each totem is actually a member of her adopted pack family). Unknown to her, she is actually a princess - her real family were victims of an evil usurper, but the assassin ordered to kill her couldn't bring himself to do it and so just left her in the forest instead (yes Snow White but with wolves instead of dwarfs).
or:
A sentient flesh golem (could maybe use the UA revenant subrace), the masterpiece of his creator. He is perfectly civilized most of the time and just wants to fit in, but when threatened (especially with fire) he is prone to enter terrifying and unnatural berserk rages.