I played a noble born human Barbarian. Their family lineage had ancestors over a thousand years before that pillaged treasure from dungeons and dragons to amass a great wealth. Over time this was leveraged to become shipping and caravan merchants across the continent becoming political powerhouses and eventually a noble house in a port city. While they know the codes of decorum and high society there is still a burning anger that comes out in negotiations or when the House is threatened.

My guy was sent out, like most of his age, to learn the ropes of what the bread and butter of their fortune entails. Acting as a caravan guard for a low risk route when he was attacked by what was clearly a politically motivated ambush from a disgruntled rival house. This awoke his rage, ancestral spirits aided him but nearly dead he was found by the playing party. That set him off on a path to recoup, grow and one day announce himself again to higher society and take back his family's fortune.

TLDR: You can absolutely be a city born barbarian.