The story would likely be full of twists. Since Kaguya was the progenitor of chakra, Naruto could have an earlier and deeper understanding of chakra manipulation. His relationship with other characters would also change. For example, his bond with Sasuke might take on a different meaning as he might have a different understanding of the Uchiha's connection to the Otsutsuki. Also, the way he approaches the concept of peace could be very different from the original Naruto, as he's been raised by someone with such a complex history.
If Naruto were raised by Kaguya, his fighting style would be completely different. He would probably be more proficient in using the more ancient forms of jutsu that Kaguya knew. His values, too, would be shaped by Kaguya. Instead of the village - first mentality that he had in the original series, he might have a more universal view of the world, seeing all living beings as part of a greater whole, similar to Kaguya's view of the world before she was corrupted by power. This would lead to a very different Naruto, and the entire Naruto - verse would be rewritten in many ways.
Well, it could be a very interesting story. Naruto might grow up with some of Kaguya's unique powers or knowledge. He could be more in tune with the Otsutsuki heritage. Maybe he'd have a different view on the ninja world, influenced by Kaguya's ancient perspective.
Well, if Naruto was raised by Kaguya in fanfiction, he'd probably be more isolated from the normal ninja community. Kaguya has her own complex history and motives. Naruto could be trained in very different combat styles, more focused on raw power and less on the teamwork - based ninja training he would have received in the original story. Also, his understanding of the tailed beasts might be completely different as Kaguya has her own connection to chakra and the world's origin.
It could be really interesting. Maybe it would involve Naruto using his talk - no - jutsu to understand Kaguya's past and change her for the better. Their relationship could bring a new kind of peace to the ninja world that was different from what Naruto achieved in the original story.
In such a fanfic, Naruto might grow up with a much stronger sense of family. Kushina could pass on her Uzumaki tenacity and unique skills directly to Naruto. He might be more confident and less of an outcast as Kushina would surely protect him from the villagers' scorn.
In such a fanfic, Naruto might learn different values from Iroh. Iroh could teach Naruto about patience and inner peace, which are things Naruto lacks in the original story. He may also introduce Naruto to different forms of strategy and wisdom, which would greatly change Naruto's way of approaching battles and life in general.
If Naruto is raised by Kaguya in fanfiction, his character could become more complex. He might inherit some of Kaguya's power and knowledge, which would change his approach to battles and relationships. For example, he could be more reserved and calculating like Kaguya, rather than the impulsive and passionate Naruto we know. This new upbringing could also isolate him from his friends at first as he might seem different.
The Naruto world would be turned upside down. With Naruto being raised by Kaguya, his understanding of chakra and the ninja way would be completely different. He might introduce new forms of jutsu based on Kaguya's ancient knowledge. The balance of power among the ninja clans would be disrupted. His upbringing by Kaguya could also lead to a change in the social structure of the ninja villages. For example, he might promote a more hierarchical system similar to what Kaguya might have been used to, instead of the more democratic - ish system that was in place in the original story.
His character development would be drastically different. He'd likely be more reserved and less of the boisterous Naruto we know. Kaguya's influence might make him more calculating and less impulsive.
Well, if Naruto and Hinata were raised together in a fanfiction, it could have a huge impact on the story. They might have a more natural progression of their relationship. Instead of Hinata just having a secret crush, they could be best friends first. Their families might also be more intertwined. Maybe they would create their own unique jutsu together, as they have spent so much time together growing up. It could also lead to some really cute childhood memories that shape their future relationship.
The plot would change a great deal. For starters, the Akatsuki might not be as big of a threat to Naruto. Since he's being raised by Kaguya, he'd likely be much more powerful from the get - go. The relationship between Konoha and Naruto would be very different. Instead of being an outcast at first, he could be seen as a potential danger because of his connection to Kaguya. Also, the search for power and the various power - up arcs in the original story might not exist in the same way. Naruto would already have access to powerful Otsutsuki techniques.
It would be a very chaotic story. Naruto's positive and determined nature would clash with Joker's insanity and Harley's wildness. Maybe Naruto would try to bring some sense of justice to their chaotic world.