Yes, the bonus damage is added to both attacks
When an effect can only happen once per turn, its statblock will specify that restriction. For example, the Assassin NPC statblock has the Sneak Attack trait (MM, p. 343; emphasis mine):
Once per turn, the assassin deals an extra 14 (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.
(In the first printing of the Monster Manual, this restriction was originally listed as in a parenthetical following the name of the trait - i.e. "Sneak Attack (1/Turn)." - but this information has since been moved into the first sentence of the description instead.)
Since the doppelganger doesn't indicate a maximum number of times it can deal the bonus damage, then it applies to all of its attacks that hit on the first round of combat against a surprised creature.