The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring?
No! The shield is not a weapon, this is designed to be a sword and board feature.
Shield is definitely not a weapon (no weapon stats), and there are currently no enchantments to make it a weapon. This is a feature designed for a brawler style (hand free for grappling) or a sword and board fighter.