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.
Harry's magic could be more about communication and unity. Since the Al Bhed have their own language and a sense of community, his spells might be focused on bringing people together or understanding different languages better. He could use magic to break language barriers, which would be very useful in the wizarding world full of different cultures and species. This would set him apart from other wizards and witches who mainly focus on offensive or defensive magic.
It could involve Harry being raised in a more diverse and unusual environment. Since the Al Bhed are a bit of an outcast group in their own world, Harry might develop a sense of being different that's even stronger than in the original story. He could use Al Bhed teachings about unity in diversity to bring together different wizarding groups against Voldemort. And his spells might be influenced by Al Bhed magic, if there is such a thing in this fanfiction, creating a whole new style of magic in the wizarding world.
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.
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.
The magic training Harry would get in this fanfiction scenario would be very distinct. Ra's al Ghul would train Harry to use magic as a weapon first and foremost. Harry would be taught spells that are not typically part of the Hogwarts curriculum. He might learn how to create magical traps, how to use magic to enhance his physical abilities like speed and strength, and how to use magic to read an opponent's weaknesses. His training would be more about survival and domination rather than the academic and sometimes whimsical magic taught at Hogwarts.
His magic might be more focused on self - defense and combat. Instead of learning spells for everyday wizarding life like in Hogwarts, Talia might encourage Harry to develop spells that can be used in a fight immediately. So his spellcasting would be more instinctive and less about following the proper incantations and wand movements precisely.
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.