Only once per attack (that hits)
The Dueling Fighting Style says:
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Damage Rolls are defined in the Combat section, under "Damage and Healing":
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.
So, the sum of all the dice that were rolled (plus the modifiers) is the "damage roll", and you add the +2 to that, not to each dice.
Note, however, that this is per attack, so if you get Extra Attack (Multiattack is a monster feature, not a PC feature) you can add +2 to the damage of each attack that hits.