If raised by Newt, Harry's ability to sense the emotions and needs of magical creatures would be enhanced. This is because Newt has a great understanding of these aspects. Harry might also develop unique magical abilities that are specific to interacting with the habitats of magical beasts, like being able to create or modify suitable environments for them with magic.
Harry's magical abilities could be more in line with nature magic. Since Newt is so involved with magical creatures and their natural habitats, Harry would probably be more attuned to the natural flow of magic. He might be able to draw power from the environment around him more easily, especially when dealing with magical creatures. His spells might also have a different 'flavor' or appearance, perhaps more in tune with the elements of nature that the magical creatures are associated with.
His magical abilities would likely develop at a much faster rate. The founders are extremely powerful and knowledgeable. They could guide Harry to discover and master magic that most wizards his age wouldn't even know exists.
Well, in such a fanfiction, Harry might grow up with a deep love and understanding of magical creatures, just like Newt. He could be more in tune with the natural magic in the world outside of Hogwarts' regular curriculum. For example, he might be able to communicate with magical beasts more easily than in the original story.
His magical abilities would likely be developed in a more archaic and powerful way. The founders had different areas of expertise, so Harry could inherit a wide range of skills. Maybe he'd be a master of defensive magic like Salazar Slytherin was known for some aspects of it, and also be proficient in charms like Helga Hufflepuff was with her kind magic.
His abilities in alchemy would be far more advanced. He'd likely be able to create potions and elixirs with ease, having learned from Flamel. Also, his understanding of magical objects, especially those related to the Philosopher's Stone, would be greater.
His magical education would likely be more hands - on. Dresden is often in the thick of things in his world, so Harry Potter would probably get real - world experience much earlier. Instead of just learning from textbooks in a classroom, he'd be out there facing actual magical threats and learning on the job.
If house - elves raised him, Harry would be extremely humble and have a strong sense of service. He wouldn't have the same pride as he does in the original series. He would also be very good at taking care of details and might be more of a behind - the - scenes helper at Hogwarts rather than the center of attention all the time.
Harry would likely be more wild in a sense. Dragons live by their instincts, and he'd pick up on that. His magical abilities could be more primal. Instead of relying on the spells taught in Hogwarts, he might use magic in a more raw and unrefined way. Also, his sense of loyalty might be more towards the dragons rather than the wizarding community initially.
His personality would probably be more aloof. Since basilisks are solitary creatures, Harry might not be as sociable as in the original story.
He would be more refined. Royal upbringing would instill in him a sense of propriety and elegance that he lacks in the original story.
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.