Protectiveness. Sasuke has seen a lot of danger in his life. He knows Sakura is a valuable person, both as a friend and as someone he cares about. So his possessiveness could stem from his desire to keep her out of harm's way. He doesn't want any bad guys to get near her.
Sasuke can be possessive of Sakura in fanfiction through his actions. He might constantly keep an eye on her during missions, not just for her safety but also because he doesn't want her attention to stray. He may also be possessive in terms of her emotions, wanting her to focus all her love and care on him alone.
One possible analysis is that in such fanfiction, the possessiveness might stem from the deep emotional bond between Sasuke and Sakura. Their relationship in the original Naruto series has many layers, and fanfiction often explores the more extreme emotions. For example, Sasuke's aloof nature might be countered by Sakura's strong will in a possessive relationship, creating a lot of interesting character dynamics and storylines.
One reason could be that Sasuke realizes his true feelings for Karin were always there. Maybe in the fanfiction, he had some sort of epiphany that his connection with Sakura was more of a convenience or misunderstanding, while his bond with Karin was deeper on an emotional level.
Some writers might want to explore different character traits. They may think that Sasuke has a more complex personality that could lead him to make such a mistake. Also, it could be a way to show Sakura's reaction in a difficult situation and how she might grow or change because of it. In addition, it might be an attempt to break the traditional 'happy ending' mold of their relationship in the original story.
One possible reason could be that Sasuke is too focused on his revenge. His single - minded pursuit of power and vengeance against Itachi leaves little room for romantic feelings. He might see a relationship with Sakura as a distraction from his goals.
One reason could be that in some fanfiction, the characterizations of Sasuke and Sakura might be off. Maybe Sasuke is portrayed as overly cruel to Sakura in a way that doesn't seem true to his character in the original series. Another reason could be that the storylines in these fanfictions are too clichéd, like always having Sasuke constantly reject Sakura in the same uninteresting ways.
In fanfiction, Sasuke might snap at Sakura due to miscommunication. Sakura could be expressing her concern for him in a way that he misinterprets. For instance, if she tells him not to go on a dangerous mission and he thinks she is trying to control him instead of just caring about his safety. Another factor could be the influence of external forces. Maybe there are other characters or events that put him in a bad mood, and Sakura just happens to be there at the wrong time, so he snaps at her.
One possible reason could be Sasuke's continued obsession with his revenge - even after they got married, his focus on past grudges might have driven a wedge between them. Another could be differences in their life goals. Sakura might have wanted a more settled family life, while Sasuke was still drawn to his wandering ways in the name of protecting the village from afar.
It could also be a way for the writer to express their own dissatisfaction with the original relationship between Sakura and Sasuke. Maybe they felt that Sakura's character was not well - developed in the context of that relationship and decided to create a more 'dramatic' storyline through cheating.
One reason could be to create drama. The idea of a relationship being broken by cheating is full of emotional turmoil which can make for an interesting read. Another reason might be some writers' preference for the Sasuke - Karin dynamic. They might see potential in their relationship that wasn't explored in the original series.