Well, it could be that Severus, out of a sense of obligation or some hidden motive, decides to take Harry as a pet. Harry, initially resistant, gradually comes to understand Severus's intentions. The story might involve Severus teaching Harry things in a very different way than he would in the normal wizard - student relationship. For example, he could be teaching Harry about defensive magic through games or challenges that are more suitable for a pet - owner dynamic. And as time passes, their relationship could evolve into something more complex and unexpected.
Perhaps in this fanfiction, Harry is actually a pet in a magical, alternate reality where Severus is a powerful wizard with a penchant for collecting unique magical creatures. Harry, in his pet form, might hold some sort of special power or significance that Severus is eager to uncover. This could lead to adventures as Severus tries to unlock the secrets within Harry while also keeping him safe from other dark wizards who might want to steal Harry for their own purposes.
Hermione would probably be one of the main characters to interact. She might be horrified at first to see Harry in such a situation and would try to find a way to free him. Ron could also be involved, perhaps being more impulsive in his attempts to rescue Harry, which might cause some problems. Dumbledore too would likely play a role. He might have some knowledge about the situation that he's not sharing right away, and his interactions with Severus would be crucial in determining Harry's fate.
Perhaps Harry takes Severus, in his pet form, on a journey through the Forbidden Forest to find a rare magical herb that could potentially break the spell. Along the way, they encounter various magical creatures and face many challenges together. Harry starts to understand Severus better through this shared adventure, even in his pet form. For example, Severus in his pet form might show his loyalty and protect Harry from a dangerous Acromantula.
In these fanfictions, Severus often develops a more nurturing side. Since he has Harry as a pet, he has to take on a caretaker role, which is different from his usual strict and aloof self. He might become more patient as he guides Harry through various situations.
It could also be a more comical plotline. Severus as a pet might constantly be getting into trouble or causing mischief in Hogwarts. Harry is left to clean up the mess and tries to teach Severus (in pet form) some manners. This could lead to a lot of humorous situations, like Severus chasing after magical creatures in the school grounds or getting into battles with other students' pets.
Maybe there's a magical spell gone wrong that links their fates to the Panthers. So they have to work together to figure out how to break the spell. Along the way, they realize they both have a deep passion for the team. They start to respect each other more as they deal with the magical and Muggle aspects related to the Panthers.
One possibility is that Harry discovers some dark secret about Severus that makes him feel he can no longer stay in the same place as him. For example, if he finds out Severus had a hand in some past event that endangered his friends. So he decides to leave to protect himself and those he cares about.
I'm not sure about a specific 'severus is harry's pet' fanfiction off the top of my head. But you can try searching on popular fanfiction websites like Archive of Our Own (AO3). There are tons of Harry Potter fanfictions there, and you might find what you're looking for by using relevant keywords in the search bar.
One interesting plotline could be Severus trying to protect Harry as his pet while hiding it from the other Hogwarts staff. For example, he might use magic to conceal Harry's presence when others are around, creating a sort of secret adventure within the castle walls.
Their relationship would become more symbiotic. Harry, as the'master', would have a sense of responsibility for Severus in his pet form. This could lead to Harry seeing Severus in a different light, less as the strict professor and more as someone vulnerable who needs protection.
A possible plot could be that Voldemort has a change of heart (maybe due to some ancient magic or prophecy) and realizes he needs Harry and Severus. Harry, on the other hand, sees an opportunity to end the cycle of violence. Severus, with his knowledge of both dark and light magic, helps them understand each other. They could then embark on a journey to unite the wizarding world, facing challenges like the resistance from the Death Eaters who don't want Voldemort to change, and the mistrust from the Order of the Phoenix who are skeptical of Harry's new alliance. Along the way, they also discover new magical abilities within themselves and each other, which helps them in their quest.