Let it Go
So, you feel your characters should behave a certain way, but everyone's having fun. I'd say let it be. Collecting trophies off victims is not out of character for a Barbarian. It doesn't seem as though the player is suffering from "My Guy Syndrome" (acting in a disruptive way and justifying it by saying "that's what my guy would do"). By the way, collecting body parts is so common a way to say "my dude's a badass" that it's a trope. It would be psychopathic to do in real life, but when a player says they want to do it in a game, I yawn a little bit.
Let the Fiction Determine NPC Reactions
But you are correct about one thing: NPCs should respond to your Barbarian's penchant for body part collection as that NPC would. That means a merchant in a civilized settlement may respond with disgust, while a bandit may be terrified by the Barbarian's prowess (probably the effect your Barbarian player is hoping for). The Barbarian may well be barred from certain settlements or even jailed. Conversely, another Barbarian, with their own body part collection, may well be impressed by your Barbarian's trophies, and may want to compare trophies.
The important thing is that you respond to the Barbarian's choices based on the story, and not in a punitive manner, e.g. "Because you are doing something I think you shouldn't do, bad stuff will happen to your character". NPCs should act based on their personalities and a common sense interpretation of the fiction.
When the Fun Stops
Now, all of this changes the moment one of your other players (or you, for that matter, you're a player too) stops having fun. Is this behavior making it difficult for players to enjoy the game? Is it making a player have an honest-to-goodness reaction that is causing them emotional distress? If so, it's time to talk about the behavior out of character, player to player, human to human.
My answer is predicated off my interpretation of your question, which is that: it's going ok, but it just seems like something I shouldn't allow, because DnD, and alignments and all that. I just don't think you should let preconceived notions of how the game is supposed to be interrupt you actual enjoyment of the game.