He would likely be more cautious and distrustful. Since he's being raised by a monster, he may not have the same sense of heroism and justice that the normal Percy has. He might see the world as a more dangerous place where only the strong survive.
Perhaps the monster is from a different pantheon and is raising Percy to bridge the gap between two different mythologies. Percy then has to deal with the cultural shock and the politics of two very different mythological worlds while being raised by this strange being.
Poseidon is a likely candidate. Since he is Percy's father in the original series, it would make sense for him to take a more active role in raising Percy. He could teach Percy all about the ocean, its secrets, and how to use his water - related powers more effectively.
Well, it could be that Percy is raised by a specific god like Poseidon but in a more direct and involved way. Maybe the god teaches him all kinds of super - secret powers and knowledge that normal demigods don't get. For example, he might be taught how to control water in ways that are far beyond what he learned at Camp Half - Blood. This could lead to him having a whole new set of adventures, facing different kinds of monsters and challenges that are related to the god's domain.
Perhaps in this fanfic, Percy is raised as a sort of prince on Olympus. He could be given tasks by the gods to prove his worthiness to be among them. This might include going on quests that are not just for the safety of the mortal world, but also for the stability of Olympus itself. He would also have to navigate the complex social hierarchies among the various gods and goddesses.
Maybe Percy would grow up with the instincts of a tiger. He could be super fast and stealthy like them. And perhaps he'd have a special bond with the tigers that would help him in battles against monsters, just like how his demigod powers usually do.
A possible new character could be an oracle - like figure who is not from the traditional Delphi but is a seer on Olympus. This character could give Percy cryptic prophecies that are related to the future of Olympus and the demigod world. There might also be a group of centaurs who are specifically trained to protect Percy and serve as his advisors, different from the ones at Camp Half - Blood.
Well, it could be about Percy learning to survive in the wild using his demigod abilities in unique ways. Maybe he discovers new mythical creatures that are only found in the wild areas he's in.
One popular 'God Percy Jackson Monster Harem Fanfic' is 'Percy Jackson's Unusual Allies'. In this fic, Percy forms unique bonds with various monsters in a way that's different from the original series. It's interesting to see how the author weaves the relationships between Percy and the monster characters. Another one could be 'Percy and the Monster Realm'. This fic delves deep into the world of monsters and how Percy navigates through it while building his harem of sorts.
One reason is the creativity it allows. In 'Percy Jackson monster fanfic', fans can give their own spin on how the monsters behave and what their motives are. Since the Percy Jackson world already has such a large and interesting cast of monsters, there are countless possibilities for new storylines. It also gives fans a chance to further explore the world of Percy Jackson. They can expand on the existing lore about the monsters, add new types of monsters, or even change the nature of the known monsters. All of this makes it really appealing to fans of the series.