Naruto's relationships would be different. He would have a father - figure in Jiraiya. Jiraiya could also introduce Naruto to new friends and allies. And Naruto would have more time to learn about his family and the secrets that Jiraiya might know about the Uzumaki clan. This could give Naruto a stronger sense of identity and purpose.
In a fanfiction where Naruto gets adopted by Jiraiya, Naruto's training would be intense and unique. Jiraiya has a lot of unorthodox methods. He could introduce Naruto to some very special and secretive forms of chakra control. Naruto might also learn more about the secrets of the Sage Mode much earlier. With Jiraiya's guidance, Naruto could form stronger bonds with other powerful ninjas that Jiraiya knew. This would give Naruto more allies and a greater understanding of the political and power dynamics in the ninja world.
His relationship with Sasuke would be different. Since Naruto would have a more stable upbringing, he might approach Sasuke with more understanding and less of the desperate need to bring him back. His friends in Konoha would also see him in a new light, as he is now under the care of two very respected shinobi. And his relationship with the village elders might be less strained, as Tsunade could act as a buffer.
If adopted by Jiraiya, Naruto would probably be more self - assured. Jiraiya would instill in him a sense of pride in his abilities. He'd also be more of a free spirit. Jiraiya's influence would mean Naruto would be more likely to take risks and explore new things without as much hesitation as in the original story. He might also be a bit more flirty, as Jiraiya has his own charm when it comes to the opposite sex.
His power level would skyrocket. He could get access to Kami's knowledge and abilities, which would make him extremely powerful in the ninja world.
His life would change drastically. He'd be surrounded by magic instead of ninjutsu. He might go to Hogwarts and have to study subjects like Transfiguration and Defence Against the Dark Arts. His daily routine would be completely different, from waking up in a dormitory to having meals in the Great Hall. He'd also have to follow the wizarding laws, which are very different from the laws in the ninja villages.
The training makes Naruto more confident in his abilities. Jiraiya constantly challenges Naruto and as Naruto overcomes these challenges, he realizes his own strength. He starts to believe in himself more, which is evident in the way he faces stronger opponents without hesitation later on. Also, Jiraiya's teachings about the ninja way and the importance of protecting those you love make Naruto more determined and focused on his goals.
Well, one possibility is that Naruto would grow up with more focused and intense training. Jiraiya could pass on his unique jutsu more directly. Maybe Naruto would learn about the world outside of Konoha in a more in - depth way earlier on, traveling with Jiraiya and having adventures that shape his understanding of the ninja world and his place in it.
Well, one possibility is that Naruto would become an even stronger ninja faster. Jiraiya could train him more intensively from a younger age. He'd also be exposed to more of Jiraiya's wild and carefree ways, which might make him more confident and a bit of a prankster like Jiraiya. And Naruto would probably have less of the loneliness he felt in the original story, since Jiraiya would be a more present father figure.
His personality could also develop in new ways. Since the RWBY world has different social structures and challenges, Naruto could become more diplomatic. He'd have to interact with different races and factions, like the Faunus. So, he might learn to be more understanding and empathetic towards those with different backgrounds, which could add new layers to his already complex personality.
Naruto's life would be completely different. He'd be part of a vampire family, which means different social norms and rules. He might not be able to be as open with his old friends as before. And he'd have to learn about vampire history and lore, which could be both fascinating and terrifying. His identity as a ninja would be in conflict with his new vampire identity in some ways, like when it comes to using violence. Vampires have their own way of dealing with threats that might not align with the ninja way.