The biggest difference would be his upbringing. In the original, he was raised by the Dursleys, which was a rather unhappy experience. With Sally Jackson, he'd have a more positive and nurturing environment. His view of magic would also be different. Instead of the somewhat formal and structured magic education at Hogwarts, he might have a more instinctive and hands - on approach influenced by Sally's parenting. And his friends would likely be different, perhaps including demigods.
In the original Harry Potter story, Harry grows up in a somewhat ordinary (albeit cruel) Muggle household and then enters the wizarding world. In a 'Harry Potter raised by Catwoman' fanfiction, he'd be raised in a world full of crime - fighting (sort of) and stealth. His moral compass would be shaped by Catwoman's actions. He might not have the same friends as in the original story. For example, Hermione might not be as interested in befriending a street - smart Harry as she was in the book - smart Harry. His relationship with magic would be different too. He might use magic more as a tool for his Catwoman - inspired activities rather than for academic or traditional wizarding purposes.
If Harry Potter was raised by Sally Jackson, it would be a wild mix of two very different fictional universes. Sally would no doubt protect Harry fiercely. He'd grow up knowing about both the wizarding world and the world of Greek gods and demigods. His character would be shaped by Sally's nurturing and her ability to handle the extraordinary. His relationship with Dumbledore might be different too. Maybe he'd be more skeptical of the wizarding world's authorities, having been raised by someone outside of that system.
She would be far more unpredictable. The original Harry was quite noble and adhered to a set of moral values. A female Harry raised by the Joker might not have such a clear - cut sense of right and wrong. She could be more likely to play pranks and use her magic in ways that are not strictly 'good'. For example, she might use magic to create chaos in a situation just for the fun of it, rather than to solve a problem in an orderly manner like the original Harry.
The original Harry Potter story is set in a world full of wizards, witches, and magical creatures like dragons and unicorns. However, in a 'harry potter is a horse fanfiction', horses would take a more central role. There could be new storylines involving horse - racing in the wizarding world, or Harry Potter using his knowledge of horses to solve mysteries. Also, his relationships with other characters might be affected by his love for horses. For instance, some friends might not understand his newfound passion, while others might join him in his horse - related adventures.
The setting would be very different. Instead of growing up in the Dursleys' dull house, he'd be in the exciting world of the circus. His friends would be circus performers rather than Hogwarts students at first. And his magic might be used in more creative and performance - based ways.
The original Harry Potter story has a very British - centric magical world. But in the 'Harry Potter raised by Pokemon' fanfiction, there could be elements from the Pokemon world which are more global in nature. For example, Harry could have Pokemon friends from different regions. His magical abilities might also be influenced by Pokemon powers. His enemies might not just be Voldemort and his Death Eaters but also those who oppose the mixing of the two worlds. And his relationships with his friends would be different as they explore this new blend of magic and Pokemon together.
In the original Harry Potter story, Harry is raised by the Dursleys, which gives him a sense of being an outcast in the Muggle world. But if raised by Frieza, he'd be in a completely different and far more dangerous environment. He wouldn't have the same kind of innocence as in the original. His moral compass would be shaped by Frieza's values, which are very different from those in the wizarding world. Also, his relationships would be different. He wouldn't have the close friendships with Ron and Hermione in the same way. Instead, he might view others as either threats or tools for his own power - seeking goals.
The most obvious difference would be his upbringing. In the original, he's raised by the Dursleys in a rather unpleasant environment. But in this fanfiction, with a scarecrow as his caregiver, he'd grow up in a rural, nature - filled setting. His values and views on magic would likely be different. He might not have the same exposure to the wizarding society's norms early on as he does in the original.
The biggest deviation would be in Harry's character. In the original, he is kind - hearted and values friendship and justice. Raised by Sephiroth, he could be more cold - hearted. Also, the power dynamics would change. Harry might not rely on the same magic sources as in the original but instead on Sephiroth - inspired powers.
The biggest difference would be his upbringing. In the original, he was raised by the Dursleys in a Muggle environment with neglect and abuse. But with Arwen, he'd be in a more nurturing and magical environment from the start. His view of magic would be different, more like the elven view which is more in harmony with nature.