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 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.
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.
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.
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.
Well, some writers might do it to explore the idea of nature vs. nurture. If Harry was Voldemort's son, how would his upbringing with the Dursleys and his experiences at Hogwarts shape him differently? It could also be a way to rewrite the entire story arc in a more subversive way. Maybe the writer wants to see how the wizarding world would react to such a revelation. And of course, for some, it's just a way to be creative and break all the rules of the original story.
Some people might create such fanfiction for the shock value. It's a very different and unexpected take on the characters. They want to see how they can twist the traditional story.
One possible reason could be that Voldemort sees Harry as a symbol of innocence that he wants to corrupt. Just like how he enjoys toying with his enemies, treating Harry like a baby might be a way to belittle him and make him more vulnerable in his eyes.
Perhaps Harry discovers some dark secret about his own past or powers that he thinks only Voldemort can help him understand. For example, if he finds out that his connection to Voldemort is more complex than he thought and that Voldemort holds the key to unlocking or controlling certain aspects of his own magic.
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 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.