He might encounter new and strange creatures. In many fanfictions, Percy could meet unique magical beings that don't exist in his own world. For example, he could come across sentient plants that can communicate or floating islands with their own mini - ecosystems.
He could meet other demigods from different pantheons. Since it's a different world, there might be demigods associated with gods that Percy has never heard of before.
Percy might meet an old enemy who has somehow been resurrected or changed in the future. This enemy could be more powerful and have a new plan to take over the world or destroy the demigods. Or he could meet a group of time - traveling demigods from an even further future who are trying to fix something in Percy's future that has gone wrong.
Percy might run into Loki. Loki is known for his trickery. He could either befriend Percy for his own devious plans or try to deceive him. Percy, being used to dealing with tricky situations in his own world, would have to be on his guard against Loki's schemes.
In a 'Percy Jackson banished to another world' fanfiction, Percy would be in for a wild ride. He could find that the laws of physics and nature are completely different. For instance, gravity might work in a strange way or time could pass at a different rate. His skills as a fighter and strategist would be crucial. He may have to learn new languages to communicate with the inhabitants of this new world. And he'd probably have to face his own inner demons while being away from his loved ones and the familiar places of his home world.
In this fanfiction scenario, Percy's adaptation to Hogwarts would be a complex process. He'd first have to deal with the cultural shock of a new magical world. The Hogwarts houses would be different from the cabins at Camp Half - Blood. He'd need to figure out which house he belongs in. His skills with a sword might not be as useful as in his own world, but he could learn to use magic to enhance his combat abilities. Also, he'd have to make new friends and enemies, and learn how to navigate the social politics of Hogwarts while still holding on to his own identity as a demigod.
One idea for a Percy Jackson famous singer fanfiction could be that a famous singer is actually a demigod in disguise. Maybe they use their music as a form of magic or to communicate with other demigods secretly. For example, their lyrics could hold hidden messages about upcoming quests or threats to the demigod community.
Some of the more traditional and rule - following demigod groups at Camp Half - Blood could be against him. They believe in following the established ways and Percy's rogue behavior goes against that. They might see him as a troublemaker and try to bring him back to the 'right' path.