In terms of chakra control, Kisame has a large amount of chakra. Naruto could learn from Kisame how to better manage and utilize his own chakra. Maybe Kisame could teach Naruto some tricks to store and release chakra more efficiently, which would enhance Naruto's combat effectiveness.
Naruto might gain a different perspective on the ninja world. Kisame has his own view as a member of the Akatsuki. Through Kisame's teachings, Naruto could understand the motives and thoughts of the 'villains' better, which could help him in future battles and negotiations.
At first, it would be full of suspicion. Naruto is a Konoha ninja and Kisame is from the Akatsuki. But as the teaching progresses, they might develop a mutual respect. Naruto would respect Kisame's power and knowledge, and Kisame might respect Naruto's determination.
In this fanfiction, perhaps Naruto is on a quest to learn more about Kisame. He could meet people who knew Kisame and through them, uncover secrets about his supposed father. This could lead to him having to make difficult decisions, such as whether to follow in Kisame's footsteps in some aspects or completely break away and forge his own path as a ninja.
The plot could involve Naruto being shunned by the Hidden Leaf Village for some reason and Kisame, seeing an opportunity, takes him in. Kisame might see Naruto as a tool at first, but as time passes, he develops a sort of father - son bond with him. Naruto would then have to deal with the fact that he's now associated with the Akatsuki, which would put him at odds with his former friends. He could use this new position to try and change the Akatsuki from within, with Kisame's begrudging support.
Well, it could be that Naruto's relationship with the other Konoha ninjas changes drastically. They might see him as an outcast because of his association with Kisame. On the other hand, Naruto might try to bridge the gap between Konoha and the Akatsuki due to his upbringing by Kisame. This would create a whole new set of conflicts and alliances within the Naruto universe.
In fanfiction, Naruto could learn various things from Mei. For example, he might learn new water - based jutsu techniques from her, as Mei is proficient in water - based ninjutsu. This could expand Naruto's combat skills and give him an edge in battles.
There could be. Fanfiction is a vast realm where all sorts of ideas are explored. However, I'm not sure if there is a highly popular or mainstream 'naruto is kisame apprentice' fanfiction. It might be a more niche concept, but it's entirely possible that some creative writer has delved into this idea to create an interesting story about Naruto learning from Kisame, perhaps incorporating Kisame's unique fighting style and view of the ninja world into Naruto's growth.
Iruka could be an important character. He's always been like a father figure to Naruto, and now with this new revelation, he might have to help Naruto deal with it. Hinata could also be in the story, and her feelings for Naruto could be complicated by his connection to Kisame. And then there could be some original characters created just for this fanfiction, like a ninja who was a former ally of Kisame and has some information for Naruto.
The plot could involve Naruto using Samehada to gain an understanding of the Akatsuki's motives. Since Samehada has a connection to the Akatsuki through Kisame, it might hold some secrets. Naruto could use it to track down other Akatsuki members or find out more about their plans.
Maybe the story focuses on Naruto learning about leadership from Hashirama. Hashirama was the first Hokage and had great leadership skills. He could teach Naruto how to make tough decisions for the good of the village, how to inspire the people, and how to deal with political situations within and outside the village.
Kisame would probably be very angry at first. Samehada is his weapon and has been with him for a long time. He might feel betrayed and try to get Samehada back by force.