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.
Well, one possibility could be that Harry grows up with a much darker and more intense family environment. Snape, being who he is, might be strict but also secretly protective. Harry could learn a lot about potions and the Dark Arts earlier than in the original story, which would shape his magical abilities in a different way.
One interesting plot could be Harry growing up with Snape's strict but ultimately caring guidance. Snape might teach Harry potions from a very young age, and Harry could develop a different kind of magic skill set compared to canon. Another plot could involve Harry having to deal with the Slytherin side of the wizarding world earlier, as Snape has strong Slytherin ties. Also, the relationship between Harry and Snape could be a complex mix of father - son dynamic and teacher - student relationship, which would be really fascinating to explore.
One possibility is that Snape would be extremely strict with Harry. He might force Harry to study Potions intensively, but also secretly protect him from the Dark Arts threats that lurk in the wizarding world. Since Snape has his own complex past and feelings towards Harry's father, this adoption could be filled with emotional turmoil for both of them.
The relationship would be complex. Snape is a complex character. He would likely struggle between his old grudges and his newfound role as Harry's father figure. He might be distant at times, but also show moments of great tenderness. For example, he might teach Harry advanced potions techniques in the dead of night, and then pretend it was no big deal the next day. Their relationship would be a slow - burning transformation from animosity to something like family.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
His character would likely become more reserved. Snape's influence would make Harry more cautious and less impulsive. He would be more attuned to the darker aspects of magic and the wizarding world.
One more to consider is 'Snape: Beyond the Surface'. In this story, the author explores Snape's relationships with other characters like Dumbledore and McGonagall. It shows how his interactions with them were not always as straightforward as they seemed in the original books and adds more depth to his character.