Since Poseidon is a god, Harry could potentially have access to a more primal and powerful form of magic. His magic might not be as bound by the rules of wizarding magic. He could draw on the vast power of the ocean, which is an almost limitless source. This could make his magic not only different in nature but also much stronger, allowing him to face more powerful enemies and challenges that a normal wizard couldn't.
Harry might have an easier time with spells related to transformation. Just as Poseidon can change the form of water, Harry could be more adept at transfiguration spells. He could also have a natural ability to breathe underwater and perform magic while submerged, which would be a huge advantage in certain situations. This would open up new possibilities for him in both the wizarding world and any underwater adventures he might have.
It could be more nature - based in a different way. The Al Bhed have their own relationship with the world around them. Harry's magic might incorporate elements from Al Bhed beliefs about nature. So instead of just using the typical Hogwarts spells for herbology, he could use Al Bhed - inspired magic that has a deeper connection to plants and the environment.
The magic would be different in that it would have a more ancient and primal feel. Since Alduin is an ancient and powerful entity, Harry's magic would be influenced by that. It wouldn't be limited to the rules and regulations of the wizarding world. He could potentially have the ability to transform into something more dragon - like, or at least gain some of the physical attributes of a dragon. His magic would be a blend of his own wizarding heritage and Alduin's otherworldly power, creating a completely new and unique form of magic that is both terrifying and awe - inspiring.
His magic would likely be more water - based. He might be able to conjure water - made weapons or shields more easily. For example, he could create a spear made of enchanted water that can pierce through dark magic defenses.
His personality would likely be more reserved. Growing up with the Greengrass family, he might not be as impulsive as in the original story. He would be taught to think before acting, a trait often associated with Slytherin. So, he would be less likely to jump into dangerous situations without a plan.
Harry would likely be more confident. Sirius is a bold and brave character, and growing up with him would instill those qualities in Harry more strongly. He wouldn't be as meek as he was in the early days at Hogwarts.
Harry would probably be more aloof. The Black family has a certain air of aristocracy. So, he might be less approachable than in the original story. He could also be more secretive, having grown up in a family where family secrets are highly guarded.
If raised by Toriel, Harry's personality would be greatly influenced. Toriel is a character who values life and peace deeply. Harry would grow up with a stronger sense of these values. He would be less likely to seek revenge or be overly aggressive in the face of his enemies. Instead, he might try to understand their motives and find more peaceful solutions. His relationships would also be more harmonious. He would be more of a mediator, using his understanding of different perspectives (thanks to Toriel's teachings) to resolve conflicts. This would make him a very different kind of hero in the wizarding world compared to the one we know from the original series.
Mentally, Harry would be more in tune with the Force. This would give him an edge in sensing danger and the intentions of others. His ability to resist the Imperius Curse might be stronger due to Jedi mental training. Instead of using spells for divination, he could use the Force to foresee events. And his telepathy - like abilities from the Force could be used in place of some magical communication spells.
If Harry was raised by the Longbottoms in fanfiction, his character would be quite different. The Longbottoms are known for their loyalty and their connection to nature. Harry would grow up with a stronger sense of family loyalty, perhaps being more protective of Neville and his family. He would also have a deeper understanding of Herbology, which could influence his problem - solving skills. Instead of being the 'Chosen One' in the typical sense, he might see himself more as part of a family unit that has a role to play in the wizarding world. His self - identity would be less centered around being an orphan and more around being a Longbottom in spirit.
His magic might develop in a more instinctive and self - taught way. Without the guidance of Hogwarts professors, Harry would have to figure out his magic on his own. He might use it more for survival and self - defense against the threats that Jason's world presents. For example, he could use magic to create barriers or traps to keep enemies at bay.