It could be that Harry has been longing for a father figure, and Snape, despite his rough exterior, has shown some moments of kindness or protection towards Harry that Harry misinterprets as fatherly love. In the potions class, something might happen that triggers Harry to call him 'dad' as an expression of that misconstrued feeling.
In the fanfiction world, the author might be exploring a different kind of relationship between Harry and Snape. Perhaps there has been some magic involved, like a spell gone wrong that makes Harry see Snape as a father figure. Or it could be a plot device to create a completely new dynamic in the Hogwarts environment, where the traditional enmity between them is turned on its head and a more family - like bond is formed, starting with that unexpected call in the potions class.
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.
One possible reaction could be shock from the other students. They know Snape as a strict and often unfriendly professor, and Harry calling him 'dad' would be completely unexpected.
One possible reason could be that in this fan fiction, Harry has developed a deep respect and a father - like bond with Snape over time. Maybe Snape has been secretly mentoring Harry in potions and has shown him kindness and understanding that Harry has never really received from his own father figure. This has led Harry to subconsciously view Snape as a father figure and the moment in potions class is just when this feeling overflows and he blurts out 'dad'.
Snape might be initially shocked. His face could turn pale and his eyes widen in disbelief. But then, a small glimmer of something like pride or warmth might flash across his face for a split second before he quickly composes himself and snaps at Harry to focus on the potion.
Maybe she's having a bad day. Everyone has off days, and Hermione is no exception. She could have had a fight with her friends earlier, or received a bad grade on a previous assignment, and that negative emotion spills over into Potions class.
In these fanfictions, Snape might ignore Harry because of his past with Harry's father. James Potter bullied Snape when they were at school, so Snape could be projecting that hatred onto Harry. It's a simple yet powerful motivation for his behavior.
Maybe he's being bullied by Snape. Snape has always been tough on Harry in potions class. His harsh words and unfair treatment could easily make Harry feel frustrated and cry.
Well, Draco could be punished by Snape for being disrespectful in Potions class. Snape might order Draco to write a long essay on the history and significance of a particular potion. This would not only be a humiliation for Draco but also a chance for him to gain more knowledge about potions. And it would add an interesting dynamic to their relationship as Draco would be resentful at first but might end up learning something valuable.
Snape could also punish Draco by deducting house points. Since Draco is in Slytherin, losing house points is a big blow to his pride and to the whole Slytherin house. This would make Draco the target of his housemates' ire, adding to the punishment's severity.
For Harry, a different potions class might make him more interested in the subject. Since he often had a tough time with Snape, a new teacher or teaching style could make him see potions in a new light. He might start to enjoy the process of making potions and even excel at it. Hermione, on the other hand, would probably adapt quickly to any changes. She might use the new opportunities in the class, like new potions to study, to expand her knowledge even further.
Perhaps it's a matter of time - sensitivity. Fury might not be able to wait until after class. There could be a time - critical situation where Peter's powers are needed immediately.