In fanfiction, they might interact more amicably from the start. Unlike in the canon where Snape was often harsh with Harry, with Female Harry Potter, he could be more understanding. For example, he might see her as a less brash and more empathetic character. Their conversations could be more open - ended and less combative.
In many fanfictions, their interaction starts with suspicion. Snape is a very guarded character, and a female Harry Potter would be an unexpected presence. So, at first, he might be cold and distant.
In slash fanfiction, Harry's impulsiveness can sometimes clash with Snape's more reserved and brooding nature. But this also creates an interesting dynamic as Harry might be the one to break through Snape's walls. For example, in some stories, Harry's curiosity about Snape leads to them having more in - depth conversations that change their relationship.
In the canon story, Snape has a very complex and often antagonistic relationship with Harry. But in fanfictions where he saves baby Harry, it's a huge departure. Here, Snape starts off as a protector, which is completely different from his role in the books where he seems to begrudgingly help Harry at times. It also changes the power dynamics as Harry may grow up with a sense of debt or gratitude towards Snape, rather than the suspicion and dislike he had in the original.
In fanfic, their relationship can be less hostile. For example, Snape may not be as harsh with Harry in class as he was in the books. He could be more understanding of Harry's situation, especially considering Harry's role in the fight against Voldemort.
In fanfiction, James Potter is sometimes portrayed as a more complex character than in the original books. He might be shown as having deeper regrets about his bullying of Snape. On the other hand, Severus Snape can be presented as even more of a tragic figure, with his love for Lily being explored in more detail. His bitterness towards James can also be magnified.
Apparition in fanfiction can also have different side - effects. While in the original series, the main concern is splinching, in fanfictions, there could be other consequences like leaving a magical trace that can be tracked by dark wizards. This adds an element of danger and intrigue to the story.
At first, Severus would probably be very suspicious of Harry. After all, Harry is an unknown figure suddenly appearing. But Harry, being the friendly person he is, would try to break the ice. He might start by complimenting Severus on his potion - making skills, which Severus is already quite good at even as a young boy. Their interaction would be a bit tense initially, but as they start to understand each other more, they could become friends or at least allies.
In some fanfictions, Snape's relationship with Harry is shown as more of a father - son relationship. For example, in 'A Different Kind of Bond', Snape takes Harry under his wing early on, and they develop a mutual respect. This is very different from the strained relationship in the books.
Mermen in fanfiction could also have different magical abilities. While in canon we don't see much of their magic, in fanfiction they could have unique spells or powers related to water. For instance, they might be able to control the weather over water bodies or create illusions within the water that can deceive even the most powerful wizards.
They could interact in a mentor - mentee kind of way in certain fanfictions. Snape, with his vast knowledge of potions and the Dark Arts, could end up teaching Ginny something valuable. This would require them to have conversations and work together, which would be an interesting dynamic considering their different houses and personalities.