One of my players is trying to say that if she smites as a paladin with the Dueling fighting style, she gets +2 to every damage die roll. This seems like a lot, as she could potentially be adding 40 damage if she multi-attacks.

Is it supposed to add 2 to every damage die roll? Or is her interpretation wrong?

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.