His status would be much different. He'd likely be seen as even more special. The other students might either be in awe or jealous. Teachers might also treat him differently, expecting more from him in terms of leadership and courage.
If Harry was King Arthur's son in a fanfiction, his life at Hogwarts would be completely transformed. For starters, he would have a new sense of self - identity. He might feel a greater responsibility towards his fellow students. His classes could take on a new meaning as he looks for connections between Arthurian magic and Hogwarts' curriculum. Also, his friends might start to see him in a new light, and he could attract new friends or enemies who are interested in his Arthurian heritage. His relationship with the Sorting Hat might also be re - examined, as it sorted him into Gryffindor based on his perceived qualities, but now there are new aspects to his character.
He'd probably have a hard time fitting in his dormitory. I mean, he might be too big for the bed! Also, Quidditch would be a whole new ball game. He could accidentally destroy the brooms or the goalposts with his strength. And the other students would likely be either in awe or terrified of him.
In this fanfiction, Harry's life at Hogwarts would be full of new discoveries. He could find secret passages that are overgrown with plants but are only visible to him because of his connection to Demeter. His friends might be a bit in awe of his new powers. He'd also have to deal with some jealousy from other students who don't have such unique abilities. And when it comes to Quidditch, he could use his nature powers to his advantage, like creating a gentle breeze to help him fly faster or making the grass on the pitch grow in a way that confounds his opponents.
His physical abilities would change drastically. He'd be much stronger and faster. No more getting winded while playing Quidditch. He could probably just fly around the pitch without a broom. And in duels, he could use his Saiyan strength to overpower opponents more easily.
In terms of his classes, Defense Against the Dark Arts would be a whole new experience. He could use his Saiyan powers to counter dark magic in ways that no one has ever seen before. He might even develop his own unique spells that are a blend of magic and Saiyan energy, making him a truly formidable wizard - Saiyan hybrid at Hogwarts.
Well, it would change a great deal. For one, he might get better grades in Potions as Snape would be more likely to give him extra help. Also, he might be more protected from the likes of Draco Malfoy and his gang, since Snape would not tolerate any bullying towards his adopted son. Harry could also have access to Snape's private library, which would expand his knowledge of magic far beyond what he could learn in the regular Hogwarts curriculum.
Their relationship might become more equal in the Hogwarts setting. In Camelot, Arthur was the king and Merlin his servant, but at Hogwarts, they'd both be students. So, they'd have to rely on each other more as equals.
His life would change drastically. He'd be in a world with different rules. No more wands perhaps, but instead ninja tools. His friends would be ninjas. He'd have to learn to move silently and quickly like a ninja.
His life would change drastically. He'd no longer be bullied by the Dursleys. Instead, he'd grow up in a more wild and free environment. He might be more in tune with his instincts.
If Harry were the son of Hela, his character would undergo a significant transformation. His entire understanding of magic and his place in the wizarding world would be shaken. He'd no longer be just the Boy Who Lived, but the son of a powerful and feared Norse goddess. This new identity would bring out a more assertive side of him. He'd be more willing to question the established rules and authorities in the wizarding world. His personality would become a blend of his original Gryffindor courage and Hela's more unyielding and powerful nature. He might also develop a new set of skills related to Hela's powers, which would change how he approaches challenges and threats in the wizarding world.
Harry's social circle would expand greatly. He'd be part of the Stark family and meet many influential people in the muggle world. At the same time, he'd still have his Hogwarts friends. This would create an interesting mix. He could introduce his wizard friends to things like video games and they could introduce him to new spells. His view of the world would be more inclusive of both magic and technology.