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.
The official ruling from Jeremy Crawford (D&D designer and rules arbiter at Wizards of the Coast) is that Dueling works with a shield:
"does the duelling [sic] fighting style work with sword and shield?"
It sure does!