Perhaps it's a way to subvert the expected narrative. In the original series, Voldemort is the ultimate villain. But in fanfiction, making him Harry's pet is a way to break all the norms. It could also be an exploration of what would happen if Voldemort was stripped of his power and had to rely on Harry in a pet - like relationship. This allows for a whole new set of character developments and storylines that deviate from the traditional Harry Potter plot.
One reason could be for comical effect. It's quite funny to think of the powerful and feared Voldemort as a pet. Another reason might be to explore a different power dynamic. If Voldemort is a pet, Harry has a certain level of control over him, which is a complete reversal of their normal relationship in the original story.
One reason could be a prophecy misinterpretation. Voldemort might initially think that Harry is key to his success in a different way than he originally thought, and so he saves him to figure out this new angle. For example, if there was a prophecy about a 'chosen one' and Voldemort realizes he needs Harry alive to fulfill some part of it that benefits him.
One reason could be to explore power imbalances. By making Harry seem like Dumbledore's pet, it shows how much power Dumbledore has over Harry's life and decisions. Another reason might be for a unique storytelling angle. It can create drama and tension, especially if Harry starts to rebel against this 'pet - like' status.
Some fanfiction writers just like to be extremely creative and break all the norms. They might think it's an interesting and unique way to explore the 'Harry Potter' universe.
Perhaps Voldemort uses magic to brainwash Harry into thinking he is his pet. Harry then has to fight against this false identity while still being under Voldemort's control. The plot could involve Harry slowly regaining his true self and using his new - found knowledge of Voldemort's inner workings (from being so close to him) to defeat him in the end. It would be a journey of self - discovery and resistance.
One reason could be a desire for revenge. Maybe in the fanfiction, Voldemort has done something even more heinous to Harry's family or friends, driving Harry to take extreme measures.
One possible plotline could be Harry being held captive by Voldemort in a way that's like a pet. Voldemort might use Harry to show his power over the Boy Who Lived. Another could be that Harry has some sort of magic that makes Voldemort keep him as a pet instead of killing him immediately.
One reason could be his pure heart. Harry's pure intentions and lack of desire for power for power's sake gives him an edge. His magic is not corrupted like Voldemort's. It flows more freely and is more in tune with the natural order of magic. So, when he casts spells, they are more effective.
In some fanfiction, Harry leaves to seek more powerful magic. He realizes that the magic in Britain is somewhat limited in the fight against Voldemort. So he travels to other places, like ancient magic - rich regions in Asia or Africa, to find a way to defeat Voldemort. He hopes that by leaving, he can gain the upper hand in the battle.
Some fanfiction writers might create this concept just for the sake of being different. They want to explore a completely new and shocking relationship that goes against the norm of the original story.
One possible plotline could be that Voldemort captures Harry early on and instead of killing him, decides to keep him as a pet. Harry, being in this strange situation, might start to develop a warped sense of loyalty to Voldemort over time.