It could be related to some prophecy or superstition within the clan. They might think Naruto is a bad omen or something that goes against their traditional beliefs, so they decide to disown him.
It could also be due to a prophecy. Suppose there's a prophecy that Naruto will bring great disaster in the future. His parents, believing in the prophecy, might decide to disown him in an attempt to avoid that future. They might think that by cutting ties with him, they can somehow change the course of events.
In such a fanfiction, the plot might involve Naruto feeling extremely alone and betrayed. He could start to question his self - worth. Maybe he'll set out on a journey to prove himself not only to the clan but to the whole village. He might find new allies outside the clan who see his potential and help him grow stronger.
One possible reason could be to seek greater power and training opportunities elsewhere. Maybe he feels restricted within his clan in terms of the type of jutsu he can learn or the level of training he can receive. So, he decides to leave to explore different ninja arts and grow stronger on his own terms.
In this fanfiction, Naruto could reject his clan because of the dark secrets or past actions of the clan. He is a character who values justice and friendship. If the clan has a history of wrongdoings that he discovers, he may not want to be associated with it. Also, Naruto has his own dreams and goals that might not fit within the framework of the clan. He has always been a bit of a rebel, and rejecting the clan could be his way of standing up for what he believes in.
One possible reason could be his status as a Jinchuriki. In the Naruto universe, being a Jinchuriki is often seen as a curse and a threat. His family might be afraid of the danger it brings to them and the village, so they disown him. Another reason could be political. Maybe there are some power - hungry individuals in the family who see Naruto as a liability to their power - grabbing plans.
Naruto could start to focus more on building relationships outside of his clan. He would seek acceptance from his friends like Sasuke, Sakura and Kakashi. Their support would give him the strength to rise above the situation. He might also start to explore different jutsu and skills on his own, without relying on any clan teachings.
Naruto's character would likely become more hardened. He would have to grow up fast and learn to rely on himself. His usual cheerful and outgoing nature might be replaced with a more reserved and determined one. He'll still have his goal of becoming Hokage, but it would now be also about proving his clan wrong.
One possible reason could be her involvement in the wizarding world. If her parents are muggles and they find out about the extreme dangers she faces in the magical realm, they might disown her out of fear for her safety. Another reason could be if she starts to show more loyalty to the wizarding community than to her muggle family, making them feel like she no longer belongs with them.
One possible reason could be that the clan fears his power. In some fanfictions, Naruto's growing strength and his potential to disrupt the clan's traditional power structure might lead to his exile. They may see him as a threat to their established ways.
In fanfiction, Naruto could be disowned by his father because of secrets. Suppose his father had some dark secrets in his past that Naruto unknowingly got involved with. His father might disown Naruto to keep those secrets buried. Or perhaps Naruto's father was under the influence of some dark magic or mind - control that made him act against his normal feelings for Naruto. It could also be a case of a prophecy that led his father to believe Naruto was a threat to the village in the future, even if there was no real evidence yet.