It could also be a part of a plot by the Malfoys to gain influence. They know Harry is a well - known figure in the wizarding world. By calling him and interacting with him, they might hope to improve their public image or gain some political leverage within the wizarding community.
In the 'Harry adopted by the Malfoys' fanfiction, one possibility is that Harry would receive a very different kind of magical education. The Malfoys might try to mold him into a Slytherin - like wizard, but Harry's own Gryffindor - like courage would constantly be at odds with that. Also, his relationship with Draco could be very complex, ranging from initial rivalry as they are brought up in the same household to a possible understanding and even friendship later on.
Some writers might do it for the shock value. It's a very unusual concept that can grab readers' attention.
Well, one possibility is that Harry would grow up with a very different set of values. The Malfoys are known for their pure - blood supremacist views. But perhaps, deep down, Harry could influence them to be a bit more accepting. He might learn dark magic quicker than in the original story, but also struggle with his own sense of right and wrong as the Malfoy values clash with what he feels inside.
It would likely be a very different story. Harry might grow up with the pure - blood supremacist views of the Malfoys. He could be trained in dark magic - adjacent skills. His relationships would be flipped, for example, he might be enemies with Ron and Hermione who are seen as 'blood traitors' by the Malfoys.
His personality would likely be a blend of his natural Gryffindor traits and the influence of the Malfoy family. He might become more guarded and calculating at times, due to the Malfoy's sly nature. But his underlying sense of justice and loyalty would still shine through. For example, he might be more hesitant to trust people initially compared to in the original story.
Maybe Harry has had some secret interactions with Snape that the readers haven't known about yet. These interactions could have made Harry feel a father - like connection to Snape, and in the heat of the moment in potions class, he just blurts it out.
The Malfoys could save Harry in fanfiction because of a personal debt. Maybe Harry once unknowingly did something that benefited the Malfoys greatly. This could be as simple as preventing a disaster from befalling their family business or saving the life of a family member secretly. When the time comes, the Malfoys feel obliged to return the favor and save Harry from whatever danger he is in. Also, it could be part of a complex plot where the Malfoys are trying to redeem themselves, and saving Harry is a big step in that direction.
In Harry Potter fanfiction, the Malfoys are often portrayed in a more complex and sometimes positive light. Some fanfictions might explore Draco Malfoy's struggle between his family's beliefs and his own emerging sense of right and wrong. He could be shown as having a change of heart, perhaps influenced by his interactions with Harry and his friends. This gives a new dimension to his character that we don't fully see in the original books.
He might become more refined in his manners. The Malfoys are a very proper pure - blood family, so Harry would likely be taught to act more like them. For example, he'd probably have better table manners and a more sophisticated way of speaking.
He might be more reserved. The Malfoy upbringing would likely make him less impulsive compared to his canon self. He could also be more calculating in his actions.