In this kind of fanfiction, perhaps Severus Snape and Tom (presumably Voldemort in his Tom Riddle form) would have a very different dynamic with Harry. Maybe they would try to shape Harry's magical abilities in their own ways. Snape might use his knowledge of potions and dark magic to train Harry, while Tom could introduce Harry to some of the more complex and dangerous aspects of magic from a different perspective.
The relationship might start off very tensely. Tom, being who he is, might see Harry as a tool or a project at first. Snape, on the other hand, may be more reluctant but still bound by some sort of obligation. As time goes on, they could develop a strange sort of co - parenting dynamic, with lots of arguments and power struggles over how to raise Harry.
It could be an orphaned wizard child from a family that was killed by Death Eaters. This would give Snape an added sense of responsibility as he might feel guilty for being associated with the Death Eaters in the past.
Snape's character could become more calculating and cold - hearted. Tom Riddle is not exactly a nurturing figure, and being raised by him would expose Snape to a world of power - seeking and self - interest. Snape may lose some of his capacity for love and affection that he had for Lily in the original story. He would be more focused on achieving power and status within Riddle's regime. His relationships with other characters would be more transactional. For instance, he would not form the same kind of complex relationship with Narcissa Malfoy as he did in the original story because his priorities would be different, being centered around Riddle's goals rather than his own personal loyalties and emotions.
Perhaps in the fanfiction, Harry adopts Tom Riddle when he's still a young boy at the orphanage. Harry then uses his own experiences of growing up without parents to bond with Tom. He teaches Tom about magic in a different way than Dumbledore did, focusing on the positive aspects of magic for good. This could lead to Tom not becoming the evil Voldemort we know.
One possible plotline could be Harry seeing the potential for good in Tom even though he knows of Voldemort's future. He might try to raise Tom with love and kindness to change his path. For example, Harry could introduce Tom to the wizarding world in a more positive light, showing him the beauty of magic for good rather than for power.
Snape's character often develops from a cold and aloof figure to a more caring and nurturing one. At first, he might be hesitant and gruff, but as he spends more time with Harry, his protective instincts kick in. He starts to show more patience and understanding, which is a big change from his usual self in the original series.
As the story progresses, their relationship could become more complex. Harry might become a father figure to Tom, teaching him right from wrong. Tom, on the other hand, could start to see Harry as someone he can rely on, but also struggle with his own inner demons that might pull him away from Harry's influence.
In such a fanfiction, one possibility is that Snape would initially struggle with his new role as an adoptive father. He has a complex personality, but over time he might start to show his protective side towards Harry and Hermione. He could teach them advanced magic, especially his own areas of expertise like Potions, in a more personalized way.
In some 'Snape Adopts Harry' fanfictions, there is a plotline where Snape and Harry have to work together to uncover a mystery at Hogwarts. This mystery could be related to the Dark Arts or some hidden secrets of the school. Along the way, their relationship evolves from a reluctant guardian - ward relationship to a more loving father - son bond. Harry might start to understand Snape's past actions and Snape, in turn, learns to open up and be more affectionate towards Harry.