He might become more reserved. Without the constant use of magic and the wizarding world around him, he could be less confident. For example, he wouldn't be able to use magic to solve problems quickly, so he'd have to rely on his wits and Muggle knowledge more, which could make him a bit more cautious.
If Harry Potter lived as a Muggle in fanfiction, his personality would transform in multiple ways. He would initially be very confused and perhaps a bit withdrawn as he tried to make sense of this new world without magic. As time passed, he might develop a stronger sense of self - reliance. He'd have to face problems head - on without the crutch of magic. His sense of justice would still be there, but it might be expressed in different ways, like standing up against bullies in a Muggle school. Also, his relationships with others would change. He'd form bonds based on different shared interests compared to his wizarding friends. Overall, his personality would be a mix of his original self and the new experiences and challenges of living as a Muggle.
His arrogance might decrease. Living as a Muggle, he'd face situations where his wizarding status doesn't matter. He might become more humble as he realizes he can't always rely on magic to solve problems.
His friends would be different. Instead of Ron and Hermione, he might meet American - born wizards with different personalities and magical abilities. They could be more into modern American culture, like pop music and movies, and bring that into their magical adventures.
He might become more reserved. Shinobi are often taught to keep their emotions in check, so Harry would be less likely to be as openly angry as he sometimes is in the original story.
His personality would be more aggressive. Yautja are hunters, and growing up with them, Harry would adopt their aggressive nature. He would be more direct in his actions and less likely to hesitate.
He would become more rebellious. Harry has always had a bit of a rebellious streak in the original series, but as a pirate, it would be amplified. He'd be more likely to question authority and do things his own way without waiting for permission.
If Harry Potter was raised by Natsu, his personality would change drastically. Natsu is a very passionate and hot - tempered character. Harry would be influenced by this passion and would be more driven in his pursuits. He would no longer be the meek boy who was bullied by the Dursleys at the start. Instead, he would be more confident and assertive. His sense of justice would also be more like Natsu's, which is more about immediate action rather than following a set of bureaucratic procedures. He would probably have a stronger sense of loyalty towards his friends as well, because Natsu values friendship above all else and would instill this value in Harry.
His personality might become more reserved. Growing up with a Shinigami would expose him to different values than the normal wizarding ones. He might not be as quick to trust or be as carefree as in the original story. Instead, he'd be more cautious and perhaps a bit more brooding, always thinking about the deeper meaning of things and the consequences of actions.
His personality would be greatly influenced by the forest environment. He would be more wild and free - spirited. Growing up away from the constraints of wizarding society, he would have a different set of values. He would be more empathetic towards animals and nature, seeing them as equals rather than something to be used. He would also be more solitary at first, but once he meets others, he would form very deep and loyal relationships as he values connection on a deeper level.
He'd probably be more timid at first. Tsuna is known for being a bit of a wimp in the beginning, and even though Harry has his own insecurities, being in a new world and a new body might make him even more hesitant. But as he gets used to things, his brave side from his past life would start to show more. He'd still have that sense of justice and a willingness to protect his friends, just like he did in the Harry Potter world.
His personality would likely become more conflicted. He has a good heart, but being around Bellatrix would make him question a lot of things. He might become more withdrawn as he tries to hide his true self from Bellatrix.