Well, perhaps in this fanfiction, Jason is possessive of Percy because he feels that Percy is the only one who truly understands him. They could have a unique connection that others don't have. Also, it might be a result of jealousy. If there are other characters showing interest in Percy, Jason's possessive nature could come out as a way to keep Percy close to him.
In the fanfiction, it could be that Jason has been through a lot with Percy. They might have shared many adventures and experiences that made Jason feel like he has a sort of claim over Percy. It could also be related to the world - building in the fanfiction. For example, if there are certain power dynamics or alliances that make Jason protective and possessive of Percy. Another aspect could be that the author wants to explore a complex relationship dynamic between the two characters, where Jason's possessiveness adds an interesting layer to the story.