His character would develop a strong sense of self - reliance. Since he has no family to fall back on, he has to depend on himself. He would be constantly on guard and more prone to taking risks because he has nothing to lose. He might also develop a darker side, as the lack of parental love could lead to feelings of anger and resentment that he has to deal with.
Harry would likely be more cynical. Without the love and support of his parents, he'd grow up feeling abandoned. He might be more distrustful of others, especially those who claim to be helping him. For example, he might not trust Dumbledore as easily as in the original story.
Harry would likely become more self - reliant. Without the support of his parents, he'd have to learn to take care of himself. He might grow more cautious and distrustful, especially if he doesn't know the real reason for the disownment.
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 would be more confident. Without the trauma of losing his parents, Harry would not have that underlying sense of insecurity. He would probably be more outgoing and less hesitant in making friends and taking risks.
He'd be more confident. With a loving mother around, he wouldn't have the insecurities that came from growing up with the Dursleys.
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.
Harry could be more distrustful of adults. Since he doesn't have a positive family figure in the orphanage, when he meets the wizarding world adults like Dumbledore, he might be more skeptical and less likely to blindly follow them. Also, his sense of loyalty might be more focused on the orphans he grew up with rather than just his Gryffindor housemates at first. He might have a harder time adjusting to the hierarchical and sometimes elitist nature of the wizarding world, coming from an environment where everyone was more or less on equal footing in terms of being an orphan.
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.