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.
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.
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.
His magic might be more plant - oriented. He could have spells that are specifically designed to make plants grow, protect them, or use them as defenses. For example, he might be able to create thick vines to block attacks or make flowers bloom to heal wounds.
Harry might have a stronger sense of self - worth. In the original story, he grew up with the Dursleys who made him feel unwanted. But if raised by a phoenix, he would be surrounded by unconditional love and protection. This could make him more confident and self - assured, not easily swayed by the opinions of others and more determined to follow his own path in the fight against evil.
Well, his story would be extremely different. For one, he'd be trained in stealth and combat skills from a young age instead of learning magic at Hogwarts in the traditional way. He might not have the same group of friends as in the original story. Instead of having Ron and Hermione as his closest allies, he could be more solitary or have assassin - type comrades. His view of good and evil would also be more blurred as an assassin, perhaps seeing the world in terms of targets and those who hire him rather than the clear - cut good vs. evil of the wizarding world. And his relationship with Voldemort would be more of a hit - or - miss rather than the epic battle of good against evil.
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.
Well, if Harry Potter was raised by Orochimaru, his magic might develop in a rather different way. Orochimaru is all about experimentation and pushing limits. He might encourage Harry to explore the darker aspects of magic earlier. Harry could end up with a more unorthodox style of spell - casting, perhaps mixing some of the dark arts with traditional wizardry. But he would also have to fight against Orochimaru's more immoral teachings to stay true to his own values.
Harry's magic could be stronger. Dorea's influence and knowledge of magic could have allowed Harry to tap into his magic potential more easily. Maybe he would be able to perform advanced spells earlier. Also, his magic might have a different 'flavor' so to speak. Since Dorea comes from a different magical background than the Dursleys, Harry's magic could be more attuned to the traditional wizarding ways of magic use.
His personality would be shaped by the strict religious environment. He'd likely be more respectful of authority figures, having been raised by nuns. In the wizarding world, this could mean he has a different relationship with teachers like Snape. He might also be more empathetic towards those who are ostracized, like the house - elves, because of the values instilled by the nuns about caring for the less fortunate. Moreover, he would be more accustomed to a simple life, so the grandeur of the wizarding world might both amaze and overwhelm him at the same time.
His magical education could be more group - oriented. Japanese culture values group harmony, so there might be more group projects and collaborative learning in magic classes compared to Hogwarts. For example, when learning complex spells, students might work in teams to master them.