If Harry was raised by Walburga Black, his relationship with Hogwarts would be drastically altered. Firstly, he might enter Hogwarts with a preconceived notion of his superiority due to his pure - blood upbringing. He could look down on some of the traditions that were inclusive of all blood types. His loyalty to Hogwarts as a whole might be divided. He would likely be more drawn to the Slytherin ideals which are more in line with Walburga's beliefs. He might not form the same close - knit group of friends from different houses. Instead, he could be more focused on building relationships with pure - blood students from Slytherin or other houses who adhered to similar values. His view of Hogwarts as a place of unity and acceptance would be clouded by the pure - blood ideology he was raised with.
His relationship with Sirius would be extremely strained. Since Walburga was Sirius' mother and they had a very difficult relationship, Sirius would likely see Harry as a product of his mother's influence. Harry, on the other hand, might be initially hostile towards Sirius due to being raised with Walburga's views.
His life would be quite different. Walburga was a strict and proud pure - blood. Harry might grow up with a stronger sense of pure - blood superiority drilled into him. He could be more proficient in dark arts - adjacent magic earlier as Walburga was from the Black family which had some connections to such magic. However, he would also likely face more internal conflict as the values of Hogwarts and his friends would clash with what Walburga taught him.
On the other hand, he could become more powerful and confident in his magic earlier. Walburga might push him hard in his magical studies, and with her knowledge of the Black family's magical heritage, Harry could develop a more assertive and perhaps even a bit of an arrogant streak in his magical abilities. But at the same time, he would also have to deal with the internal conflict of the Black family's dark past.
It would be a very different relationship. He might see Hogwarts as a foreign place at first. He'd be used to Odin's teachings, which are very different from Hogwarts' curriculum.
Harry being raised by Walburga Black would have a profound impact on his personality. Since Walburga was a firm believer in pure - blood supremacy and had a rather tyrannical nature, Harry would grow up in an environment full of rules and expectations based on blood status. He might become more arrogant towards those not of pure - blood heritage. His sense of humor, which was a big part of his canon personality, might be stifled. He could be more focused on upholding the family's 'proud' pure - blood name and less interested in making friends with Muggle - borns or half - bloods. However, as he got older, he might also start to rebel against the values he was raised with, much like Sirius did against his family.
Well, it could potentially make their relationship less hostile from the start. Harry might see Snape in a more positive light, as Snape is doing something kind for him. Snape, on the other hand, might also feel a bit of a connection to Harry, knowing he's starting this important journey.
If Harry was raised by Bellatrix, his character would likely be much darker. Bellatrix is a Death Eater, so Harry would be exposed to dark magic and Voldemort's ideology from a young age. He might be more ruthless and less empathetic compared to the Harry we know. For example, he might not value friendship as highly since Bellatrix doesn't really understand the concept of true friendship. Also, his view of the wizarding world would be skewed towards the pure - blood supremacist ideas that Bellatrix adheres to.
Their relationship could grow deeper as they share the responsibility of shaping the next generation of wizards. They would have to communicate constantly about the students' progress, and this could lead to a better understanding of each other's thoughts and feelings. For example, Hermione might learn more about Harry's intuition when it comes to students with potential, and Harry could gain more knowledge from Hermione about the importance of academic rigor.
They could form a sort of alliance. Harry, being the hero figure, could bring Draco into his circle of friends more often. Draco, with his Slytherin knowledge and connections, could help Harry in understanding some of the more mysterious aspects of Hogwarts, like the secrets of the Slytherin common room. Their relationship could bridge the gap between Gryffindor and Slytherin, changing the whole dynamic of the houses at Hogwarts.
His Hogwarts experience would likely be very different. For one, he might be more physically fit and confident in combat situations. He could be less intimidated by the likes of Snape or Malfoy, as the Dragon Slayer would have taught him to stand up for himself. Also, he might introduce some new skills to Defense Against the Dark Arts, like dragon - slaying techniques adapted for magical creatures.