Harry's character would develop in a very distinct way. Since Sally Jackson is from a world of Greek mythology, Harry would be exposed to different values. He might become more of a team player in a broader sense, not just within the Gryffindor house. He would also likely be more adaptable, having to deal with the blend of two very different magical worlds. His sense of identity would be more complex as he tries to figure out his place in both the wizarding world and the world related to Greek gods.
He'd be more confident. With Sally's love and support, he wouldn't have the self - doubt that he had in the original story. He'd be quicker to trust people and might be more open - minded about different forms of magic.
His magical abilities might develop in a different way. The Malfoys could introduce him to different types of magic or magical artifacts. Maybe he'd learn more about the Dark Arts in a different context, not for evil purposes but just to understand it better. He could also become more skilled in areas like Occlumency earlier on, since the Malfoys might see it as an important skill to protect their family secrets. His relationship with his Gryffindor traits would be tested. He might still have his bravery, but it could be expressed in different ways, like standing up to the Malfoy family's wrongdoings.
If Sally Jackson adopts Harry Potter in fanfiction, it could be really interesting. Sally is such a loving and strong mother figure. Harry would probably have a more stable home life compared to what he had with the Dursleys. He might also get to know Percy Jackson and they could share their magical experiences. And who knows, he could end up being part of some epic quests involving both the wizarding world and the world of Greek mythology.
He'd likely be more inquisitive about different forms of magic. Instead of just relying on the spells taught at Hogwarts, he'd constantly seek out new magus - based knowledge.
Harry would likely become more feared by some in the wizarding world. Since the Basilisk is often seen as a terrifying and dangerous creature, his connection to it might make others view him with suspicion. He would also have to grapple with the darker side of magic that the Basilisk represents, which could make him more brooding and introspective.
He'd be more confident. With a loving mother around, he wouldn't have the insecurities that came from growing up with the Dursleys.
Harry would be more open to dark magic. Since Voldemort is his adoptive father, he would be exposed to different aspects of magic earlier. He might not have the same fear of the dark side as in the original story. For example, he could be more confident in using spells that are considered dark but for good reasons, like using the Imperius Curse to stop a criminal.
He could develop a more cunning side. Bellatrix is a sly character, and Harry would need to learn to be crafty to survive in her world. This would be a big change from his usually straightforward and brave self. For example, he might learn how to deceive Death Eaters or other threats.
Harry would be more confident in his physical abilities. Since Raiden is a powerful fighter, he'd train Harry to be strong and fearless in combat. This would be different from the Harry we know who often relied on his wits and magic in dangerous situations. Harry might also be more stoic, like Raiden, and not show his emotions as easily.
Harry would be more daring. Jack Sparrow's influence would make him less afraid of taking risks. In the wizarding world, this could mean he'd be quicker to explore dark and dangerous places. He might not wait for permission or for a group to support him, but rather just dive in like Jack does when facing a new situation on the sea.
Harry's character would develop a sense of divinity within him. Since he's raised by Greek gods, he'd have a different relationship with power. He wouldn't fear power as much as he might in the original story. Instead, he'd see it as a natural part of his being. He'd be more in tune with nature and the elements, like the gods are. His relationships with other characters would also change. He might be more aloof towards some of his Hogwarts friends at first, as he'd have this whole other world of experiences from his upbringing among the gods that they couldn't understand right away.