Perhaps he likes to see how fans re - imagine his character. He might be curious about how others view his actions and motives throughout the story.
It could be that he enjoys the creativity in fanfiction. The different plotlines and character developments that fans come up with might be appealing to him. For example, he could be interested in how fans might rewrite his relationship with Harry in a more positive light or explore the what - ifs of certain situations in the wizarding world.
One reason could be that in this fanfiction, Snape has some hidden kindness towards Harry that he doesn't show in the original story. Maybe he secretly protects Harry from other threats, which makes Harry feel safe around him.
One reason might be that the fanfiction doesn't capture the essence of Mcgonagall as Harry knows her. He has a deep understanding of her from their time at Hogwarts, and if the fanfiction misses that, he won't like it.
One reason could be that he doesn't like others taking creative liberties with his story. He might feel that it's his story and only the original version should be the 'true' one.
One possible reason could be copyright infringement. If the fanfiction uses too much of the original 'Harry Potter' material without proper authorization, J.K. Rowling (or those in charge of the franchise) might take legal action. Another reason could be if the fanfiction misrepresents the characters in a way that goes against the intended spirit of the original work.
One possible reason could be that Harry breaks a major rule in the wizarding world within the fanfic. Snape, being a strict and by - the - book person, grounds him as a form of punishment. For example, if Harry uses dark magic without permission, Snape would surely take strict action.
It could be that in fanfiction, Snape has been through some additional trauma or experiences that change his perception of Lily. For instance, if he discovers something about Lily that he doesn't like, such as her involvement in some secret or her having a different side to her character that he finds unappealing. This would cause him to dislike her, as opposed to his more nuanced feelings in the original story.
One reason could be that Harry sees Snape's Patronus, which is the same as his mother's. This could be a strong indication of Snape's true feelings and make Harry trust him.
Maybe Harry Potter is tired of the unrealistic portrayals of Snape in fanfiction. Often fanfictions can exaggerate characters' traits, and Harry, who knew Snape in real life, might find these interpretations too far - fetched.
One reason could be a misfired spell by another character. For example, if a young wizard was experimenting with a transportation spell and it backfired, it could accidentally force Harry to be attached to Snape in this 'ride' situation. Another reason might be a plot by a dark wizard who wants to see chaos at Hogwarts. They figure that forcing Harry and Snape together in this strange way would disrupt the normal order of things.
In some fanfictions, dark magic interference might cause a wand to explode. If a dark wizard tries to curse a wand or disrupt its magic, it could lead to a catastrophic failure. This could be part of a plot where a hero's wand is targeted by an enemy.