Well, Gojo could be a super powerful outsider. Maybe he comes to the Naruto world to study its unique chakra system. His Infinity ability would be a game - changer. He could use it to easily deflect most ninjutsu attacks. And his Six Eyes could help him detect hidden enemies or traps that normal shinobi might miss.
Well, one similarity could be their determination. In the fanfic, like in the original series, both Naruto and Kakashi are very determined. Naruto is always striving to become Hokage and Kakashi has his own goals and missions he's dedicated to. They also might share similar combat styles in the fanfic, perhaps Naruto starts to adopt some of Kakashi's strategic thinking during battles.
He might be shocked. In the fanfic, if he sees things like his friends' future achievements or the new threats that emerge, it would surely startle him. For example, if he discovers that one of his friends has a family of their own and has become a great leader in the village, he would be proud but also a bit surprised at how much has changed.
Sasuke could be a popular character. He might be involved in the story in a way that either helps or hinders the situation where Naruto hurts Kakashi. For example, if Sasuke tries to stop Naruto from doing something stupid that would hurt Kakashi.
Another reason could be related to Kakashi's past actions. If there are some secrets or things Kakashi did in the past that come to light and Naruto doesn't like them, he could snap at Kakashi. Naruto has a strong sense of justice and loyalty, so if he feels betrayed or misled by Kakashi, his reaction could be to snap.
With his usual determination. Naruto has always been one to face difficulties head - on. So, even if he's shocked by what he sees in the fanfic's future, he'll likely try to find solutions immediately.
Well, there could be. Fanfiction writers love to explore unique and unexpected relationships. Since Naruto and Kakashi are both popular characters in the Naruto universe, some writers might come up with such a plot for the sake of creativity and exploring different storylines. But it's not a part of the official Naruto story.
I recommend "I am the Second Young Master of the Flagwood Family" to you. The main character of this novel, Flagwood Miren, transmigrated to the Naruto World and became the second young master of the Flagwood Family. He is the same age as Kakashi. The plot of this novel also included other Naruto characters, such as Haki Shuogu, Ban, and Bo Fengshui Gate. I hope you like this fairy's recommendation. Muah ~😗
Kakashi might be initially surprised. But then, he would probably use it as a teaching moment for Naruto. He could say something like 'Naruto, stealing is not the way of a ninja.' And then give him a punishment like extra training.
Toge Inumaki is important too. His special cursed speech ability makes him stand out. Also, the principal of the school, Masamichi Yaga, is likely to be important as he is in a position of authority and has an influence on Gojo and the other students. There could be other characters like the members of different clans that also play roles in the complex web of the story.