Another possible main character could be Dudley Dursley. He might interact with the pet in a different way compared to the others. Maybe he is jealous of the attention the pet gets from his parents or he could use the pet to bully Harry in some new and creative ways in the context of the fanfiction.
Petunia Dursley could also be a main character. She is in charge of the household and the pet. In the fanfiction, we could explore how she views the pet and how her attitude towards the pet changes when strange things start to happen, perhaps due to Harry's influence or the pet's own mystery.
It could be a young wizard who is an outcast in the wizarding world. Maybe they have some unique powers that others fear, so they choose to hide as a pet to avoid persecution.
One possible storyline could be that the Dursleys' pet, perhaps their cat, discovers Harry's wizarding abilities before the Dursleys do. It might then have some comical or tense interactions with Harry, like trying to stop him from using magic in the house for fear of getting in trouble itself.
Perhaps there's a mentor figure, an old and experienced pet trainer who gives Harry advice from time to time. Another possible character could be Harry's rival in the pet training world. This rival might always be trying to outdo Harry in training different pets, which adds some drama to the story.
Maybe it's an anonymous gift from someone who has been observing Harry. This person could be a former student of Hogwarts who has a special connection to snakes and sees that Harry has a kind heart and would take good care of the snake. So they leave the snake for Harry to find with a simple note that just says 'For Harry Potter'.
Perhaps an ancient evil that has been awakened by the presence of the pet dog in the wizarding world. This entity doesn't like the idea of a human - dog bond being so strong in a place full of magic. It starts causing chaos in Hogwarts and the surrounding areas, and Harry and his pet dog have to figure out how to stop it.
One main character could be a young, aspiring wizard who has always dreamed of having a magical pet. He visits the pet shop regularly to look at all the amazing creatures. Another could be the eccentric old owner of the pet shop who has a wealth of knowledge about magical pets.
Maybe Professor McGonagall. She was strict but also had a soft spot for Harry. She might take Harry under her wing more in this fanfiction and make him her 'pet' student. She could teach him advanced Transfiguration skills or involve him in Gryffindor - related leadership tasks that give him an edge over the other students.
Harry Potter himself could be a main character. He might be on a mission to find a pet for a friend or to solve a mystery related to the pet shop. Hermione would also be a great choice as she's always interested in learning about new magical creatures. And Hagrid, of course, with his love for all beasts, could play a major role.
It could be a snowy owl, just like Hedwig. In the original series, Hedwig was Harry's loyal and beautiful owl. A fanfiction might follow this idea and have Harry find a young snowy owl that he bonds with immediately.
Hermione could be a main character. She is always by Harry's side and her intelligence would be crucial in any discovery. She might be the one to figure out the clues or research the background of the discovery.