If Naruto was adopted by Tsume in fanfiction, it would be really interesting. Tsume is a motherly figure in her own way. She could provide Naruto with the love and support he lacked growing up. He might be trained more rigorously in the ways of the Inuzuka clan, which emphasizes a close bond with their ninja dogs. This could lead to Naruto having some amazing combinations of his own Uzumaki abilities and the Inuzuka clan's skills. His relationship with other characters, especially those in the Inuzuka clan, would be a major aspect of the fanfiction. He could form strong bonds with the younger Inuzuka members and become like an older brother figure to them.
One possible plot could be that Naruto gains a new family structure. Tsume, being a tough and no - nonsense character, could instill in Naruto a sense of discipline and loyalty similar to what she teaches her own clan. Naruto might learn new jutsu related to the Inuzuka clan, like how to work better with ninja dogs. This would change his combat style and make him more versatile in battles.
Another plot twist could be that Tsume was actually chosen by a mysterious force to adopt Naruto for a specific mission. She initially doesn't know about this and raises Naruto out of kindness. But as Naruto grows older, they start to uncover clues about this mission. It could be something that has the potential to change the entire shinobi world, and they have to face many challenges and make difficult decisions along the way.
Naruto might develop a more ferocious side to his personality. Since Tsume is from a clan known for their wild and somewhat aggressive nature, Naruto could pick up some of those traits. He might be more direct in his approach to problems and less likely to hold back in a fight.
Naruto's character would become more disciplined. Tsume is a strict character, and under her care, Naruto would have to follow rules more closely. His impulsiveness might decrease.
Naruto's life would be completely different. Tsume is a strict and proud character. Under her care, Naruto would likely be pushed to his limits in training. He would also have a different living situation. Instead of living alone in his small apartment, he'd be in the Inuzuka compound. He might be given a ninja dog of his own early on, which would change his combat style and how he approaches missions.
In this fanfiction, Naruto might save Tsume from a powerful enemy. Maybe he uses his Rasengan to defeat the foe who has Tsume cornered. His quick thinking and combat skills could be the key to her rescue.
In such a fanfiction, Naruto might teach Tsume new ninjutsu skills. For example, he could share his unique Rasengan technique, explaining the chakra control required step by step.
The social status in the village could also be a problem. Naruto is a hero to the village, but he also has his share of enemies. Tsume, on the other hand, has her own standing in the village. Their relationship could make them targets for those who oppose Naruto or who don't like the idea of an Inuzuka getting involved with an Uzumaki. Also, they would need to balance their duties as shinobi with their relationship, which can be really tough.
It's likely a mentorship relationship. Naruto, as Tsume's master, would be in a position of authority and guidance. He'd be responsible for Tsume's growth as a ninja.