One reason could be jealousy. If in the fanfiction Snape is getting more attention or praise from the wizarding world than Lily, she might start to dislike him. Another reason could be that Snape has a different view on blood purity, and Lily strongly disagrees with him, which leads to her hatred.
It's possible that in the fanfiction, Lily's character is developed in a way that she has zero tolerance for Snape's rudeness or his cold - heartedness. If Snape is constantly mean to others in the fanfic, especially those Lily cares about, she would naturally grow to hate him.
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.
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.
It could also be due to some external factors. Maybe there are threats from dark forces that he doesn't want to involve Lily in. If he is getting deeper into the dark arts or some dangerous situations, he might choose to give up on her to protect her from the potential harm that could come her way because of his associations.
Well, in some fanfictions, it could be because Snape's actions towards her friends in the original story. Maybe the fanfic writer wants to explore a different side of their relationship where Lily has had enough of Snape's dark arts fascination or his association with the wrong crowd at Hogwarts.
Perhaps in this fanfic, Harry has been influenced by dark magic or some evil entity. This external influence could distort his perception of Lily and turn his normal feelings for her into hatred. Dark magic has the power to corrupt one's mind and change relationships in unexpected ways.
One reason could be jealousy. Snape might be jealous of James' popularity and charm. James, on the other hand, could be jealous of Snape's close friendship with Lily in their early years. Another reason is their different personalities. James was outgoing and bold, while Snape was more introverted and brooding. Lily might be attracted to different aspects of each of them, which could lead to the fight as they both compete for her attention.
It might be because Snape said or did something hurtful to Lily's friends. Lily is a loyal person, and if Snape insulted or harmed her friends in some way, she would be furious enough to send a Howler. In the context of fanfiction, there could be a new character introduced who Snape treats badly, and this angers Lily.
One reason could be that some fans are very purist about the original Naruto story. They might feel that fanfiction often distorts the characters or the plot in ways that go against the essence of what Naruto is all about. For example, changing a character's personality completely to fit a romantic or dark and edgy narrative that doesn't align with the source material.
Perhaps in the fanfiction, James is seen as more accepting of Lily's friends and her social circle. Snape, on the other hand, might be associated with a darker crowd or have some beliefs that Lily doesn't agree with. For example, if Snape is too involved with the Dark Arts in the fanfiction, it could be a turn - off for Lily.
One possible theme is unrequited love. Snape loved Lily deeply, and their Patronuses, both being doves perhaps in some fanfictions, could symbolize that love. His Patronus could be a constant reminder of his love for her, even though she was with someone else.