There are several possible consequences. Percy may find himself isolated as other characters might be afraid to get too close to him due to Zeus's possessiveness. His growth as a character could be stunted as Zeus tries to mold him according to his own desires. It could also spark a rebellion among the demigods who don't want Zeus interfering so much with one of their own. Moreover, the relationship between the gods could become more complex and tense, especially between Zeus and those who support Percy's independence, like Poseidon and Athena.
The consequences can be quite significant. For starters, Percy's relationships with his friends could be strained. His friends might be worried about Zeus's influence over him. Percy himself could start to resent Zeus's possessiveness, which might lead to him acting out. In the larger context of the fanfic's world, it could also cause an imbalance in the power dynamics among the gods. If Zeus is too focused on Percy, other important matters in the divine realm could be neglected, leading to chaos.
In some fanfics, Zeus might be possessive of Percy because Percy represents a powerful force or potential. Zeus, being the king of the gods, wants to control all sources of power. Percy's abilities and his role in the demigod world could make Zeus see him as something to be owned or at least controlled.
In such a fanfic, one possible consequence could be a major upheaval in the world of the story. The child would be of divine and demigod lineage, which might attract all kinds of unwanted attention from other gods and mythical beings. It could also lead to a power struggle as different factions might try to use the child for their own purposes.
One consequence could be Percy losing his human connections. His relationships with his friends and family were based on his shared humanity. As a god, he might find it hard to relate to them in the same way. For example, his mortal friends may age and die while he remains an immortal god, leading to a sense of isolation.
Love could be the main reason. Since he loves Annabeth deeply, he doesn't want to lose her, so he becomes possessive.
For the demigods, things could change a lot too. Percy is a champion of the demigods, and his victory over Zeus might mean better treatment for demigods. They could be more respected and have more rights in the world of gods and mortals. However, it could also make them targets of those who are against Percy's actions.
One possible consequence is that it could lead to a split in their group. If they are part of a team or a community, their animosity could cause others to take sides, creating factions.
She might start to compare the real Percy (if he exists in the same universe) to the fan - created versions. This could lead to misunderstandings or unrealistic expectations.
Piper could become so engrossed in the fanfiction that she neglects her own real - life relationships and responsibilities. For example, she might skip out on meetings with friends or not do her schoolwork (assuming she's a student) because she's too busy reading Percy fanfiction.
One consequence could be that his enemies might see this as an opportunity to attack him further while he's weak. But on the positive side, this could also lead to new alliances being formed in the fanfiction world. Other characters who previously had no connection with Percy might come forward to protect him, seeing his situation as unjust.
It could also be about power. Percy might realize that with Zeus out of the way, he can reshape the world of the gods in a more just and fair way. He could believe that Zeus's rule is corrupt and that killing him is the only way to bring about real change.