Another reason could be related to his personal growth. He might be trying to master the power of the eye in private. Naruto has always been one to face challenges alone at first. By hiding his eye, he can focus on understanding and using its power without distractions or the pressure of others' expectations.
Maybe he's trying to hide his identity for a secret mission. In some fanfictions, Naruto might be undercover, and covering his face is a simple way to avoid being recognized.
Well, it could be that Kakashi, being a character who values the integrity of the original Naruto narrative, sees fanfiction as a deviation. Fanfiction might include out - of - character scenarios for him or the other characters. He might also dislike the fact that some fanfictions can be of very poor quality in terms of writing and plot, which could be a reason for his dislike.
Maybe he thinks fanfic often distorts the real story or characters of Naruto. In the original Naruto series, Kakashi has his own understanding and relationship with Naruto. Fanfic might create some scenarios that go against his view of Naruto or the world they live in.