Ja, det gör

När en effekt bara kan ske en gång per tur anger Monster Manual det. Till exempel har mördaren (MM 343) Sneak Attack:

Sneak Attack (1/Turn). The assassin deals an extra 13 (4d6) damage when it hits a target with a weapon attack and has advantage on the attack roll, or when the target is within 5 feet of an ally of the assassin that isn't incapacitated and the assassin doesn't have disadvantage on the attack roll.

Eftersom doppelgangaren inte anger ett maximalt antal gånger kan det hantera bonusskadorna, så gäller det för alla attacker.