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 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.
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.
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.
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.
Well, 'Secret Encounters' is a Severus and Harry lemon fanfiction that many fans seem to like. It has a good balance between the emotional connection and the more adult - themed interactions. But be aware that this kind of fanfiction isn't for everyone as it can be quite explicit in nature. There's also 'Forbidden Love's Flame' which takes their relationship to a new level of passion.
Lily Potter is often a significant character. Since she was friends with Severus in the past and is Harry's mother, she can play a crucial role in fanfictions. Her relationship with Severus and how it affects Harry can be explored in different ways, like through memories or magical connections.