In fanfiction, it could also be that Kakashi has seen Naruto's potential. He knows that Naruto has the ability to become a great ninja and change the ninja world for the better. Kakashi wants to protect that potential. If Naruto were to be killed or severely damaged early on, all the great things he could achieve would be lost. So, Kakashi is protective to ensure that Naruto can reach his full potential and fulfill his destiny as a ninja.
One reason could be that he sees Naruto as a younger version of himself in some ways. Both have had their share of loneliness and challenges growing up. Kakashi may want to prevent Naruto from going through the same pain he did. Another reason is Naruto's importance to the village. As the jinchuriki of the Nine - Tails, Naruto is a key asset. Kakashi, being a loyal ninja, wants to protect the village's future which is tied to Naruto.
Kakashi might also be protective because of his relationship with Naruto's father, Minato. Minato was Kakashi's teacher and Kakashi respected him a great deal. Since Naruto is Minato's son, Kakashi feels a sense of obligation to look after him. Additionally, Naruto's determination and good heart have likely won Kakashi over. Naruto's unwavering belief in his friends and his goals makes Kakashi want to ensure that such a pure - hearted person doesn't get crushed by the cruel world of ninjas.
One great fanfiction is 'Kakashi's Unwavering Shield'. In this story, Kakashi goes to extreme lengths to protect Naruto from various threats within the village. He even stands up against the council when they try to make unfair decisions regarding Naruto. It really shows the depth of his protectiveness.
One possible reason could be that Kakashi is preoccupied with other important matters in the village. Maybe he has some top - secret missions or tasks assigned by the Hokage that demand his full attention, leaving him with little time to focus on Naruto.
One possible reason could be that Naruto might feel Kakashi is too strict with his training. Maybe Kakashi's methods are too harsh for Naruto, making him frustrated and gradually developing dislike.
One possible reason could be Kakashi's reputation as a strict and powerful teacher. Naruto might be in awe of his skills and fear making mistakes in front of him.
One possible reason could be that Naruto might be trying to prove his independence. He has grown and wants to show that he can handle things on his own without always relying on Kakashi's guidance.
One possible reason could be that Kakashi might misjudge Naruto's abilities in the fanfiction. Maybe Naruto shows some new or unorthodox skills that Kakashi doesn't understand at first and thus rejects him. Another reason could be a misunderstanding. For example, Naruto might be involved in something that looks bad from Kakashi's perspective, but is actually a misunderstanding.
One possible reason could be that in the fanfiction, Kakashi might have made a decision during a mission that Naruto strongly disagreed with. For example, Kakashi could have chosen to sacrifice something important to Naruto in order to achieve the overall mission objective.
Maybe Naruto admires Kakashi's skills. Kakashi is a highly respected ninja, and Naruto might want to be as good as him. So he starts acting like him to learn from him.
One reason could be that the fanfiction changes Kakashi's teaching methods too much. Naruto is used to Kakashi's unique style in the original story, and if it's drastically altered in a way that doesn't seem right to Naruto, he might not like it.
Perhaps Anko has a personal connection to Naruto. Maybe she sees a bit of herself in him, like his determination in the face of difficulties, and this makes her feel a sense of kinship and thus the need to protect him.