He might become more sneaky and calculating. Since he's a thief now, he would have to be good at hiding his actions and planning his 'heists'. His moral compass might also shift a bit as stealing isn't exactly an ethical behavior in the normal wizarding world.
The perception of Harry would be turned upside down. In the original series, he is the symbol of good and the one who defeats Voldemort. But in this fanfiction, he is the one causing harm, which makes readers question everything they thought they knew about his character. His motives, actions, and relationships would all be seen in a new and disturbing light.
He becomes more assertive. Instead of waiting for others to tell him what to do, he takes the initiative. For instance, he would be more likely to confront the Ministry of Magic directly about their wrongdoings.
He might become more in tune with nature. Since the Phoenix is a symbol of rebirth and nature in the wizarding world, Harry could develop a stronger sense of environmental magic. For example, he could be able to communicate with other magical creatures more easily, not just the Phoenix.
He becomes more humble. Elves are used to serving and being in a subservient position. Growing up with them, Harry would learn the value of every being, no matter their status. He'd be less likely to show off his magical abilities like he might have in the original story.
He becomes more brooding. The knowledge of necromancy and the power it brings can isolate him as he realizes it's a power not many understand or accept. So he might withdraw from his usual jovial self.
In 'Harry Potter son of Loki' fanfiction, Harry changes significantly. His sense of identity is greatly shaken. He no longer just sees himself as a simple wizard. He now has this connection to Norse mythology. This makes him more curious about different forms of magic and otherworldly things. He may become more of an outcast as well, since his heritage is so different. But this also makes him stronger in some ways. He has to rely on himself more and develop his own unique set of skills and powers that come from both his Potter and Loki lineage.
Harry in this fanfiction is likely to be more ruthless. Voldemort's upbringing would make him less empathetic. He may value power above all else and be more willing to use dark magic to achieve his goals. His relationships would also be different. He wouldn't have the same close - knit group of friends as in the original story. Instead, he might be surrounded by Death Eaters or those who follow Voldemort's ideology.
In this fanfiction, Harry's change is quite significant. Firstly, his view of himself and his place in the world is completely altered. Instead of just being a wizard, he is now part of something much bigger. This new identity gives him a different kind of courage. He's not just fighting for the wizarding world against Voldemort, but also has a connection to Asgard. His relationships with his friends also change. They might see him in a new light, and he might have to deal with their different reactions. He could become more of a leader, using his new powers and status to inspire others. And his magic might even be influenced by his Asgardian heritage, making it more powerful or having different manifestations.
In this kind of fanfiction, Harry often becomes more proficient in potions and other areas that Snape is good at. His relationship with Snape also makes him more empathetic towards Snape's past and the Slytherin house in general. He may not be as quick to judge people based on their house. Also, he might develop a more complex view of the wizarding world's politics, as Snape is deeply involved in some of the darker aspects of it.
His character might become more confident. Since he is one with magic, he has a newfound sense of power and purpose.