One possible reason could be that fanfiction often takes creative liberties that may distort the complex character of Snape that J.K. Rowling created. Snape has a very intricate backstory and personality in the original series, and fanfiction might simplify or misinterpret it.
One possible reason could be his lingering resentment towards James Potter, Harry's father. Snape and James had a long - standing feud during their school days. Since Harry looks a lot like James, Snape might be transferring his old grudges onto Harry.
It can create a lot of tension. If Snape is constantly hating in the fanfiction, it can make the atmosphere very strained. For example, if the story is set in Hogwarts, his hatred might spill over into his interactions with students and other teachers, making the whole school environment seem more hostile.
One reason could be that he realizes Harry has a lot of potential and wants to nurture it. He sees that Harry is in a difficult position, being the target of Voldemort, and decides to help him despite their past differences.
One possible reason could be Snape's past experiences. Maybe he had a difficult childhood himself and doesn't know how to handle having a son. He might project his own insecurities and traumas onto Harry. Another reason could be that he sees too much of Harry's father in him. Since Snape had a long - standing feud with Harry's father, seeing those traits in Harry could trigger his hatred.
One reason could be to add a shock factor. It completely changes the expected family relations in the original story. Another reason might be to explore Snape's character more deeply. If Harry is his son, it gives a new perspective on Snape's actions and emotions towards Harry throughout the series.
In fanfiction, anything can be reimagined. It could be possible through creative plot devices like a hidden relationship between Snape and Lily Potter before she was with James. Maybe there was a secret encounter that led to Harry's conception, and the truth was hidden all along.
One possible reason could be that Snape is secretly trying to protect Harry. Maybe the potion has some sort of protective charm or antidote against a dark magic threat that Snape has foreseen. He may not show it openly, but his actions are driven by his long - standing, complex feelings towards Harry's mother and his sense of duty.
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 reason could be jealousy. Maybe Snape gets jealous of Lily's other relationships or her popularity at Hogwarts. Another could be a fundamental difference in their beliefs or values that grows over time into hatred.
Isolation. Snape is a lonely character. He doesn't have many friends, and he is constantly in a difficult position between the Death Eaters and the Order of the Phoenix. His feeling of being alone and not being fully understood by anyone can lead to crying in fanfiction.