In a 'Harry is Bellatrix's pet' fanfiction, the plot could begin with Bellatrix seeing Harry as a unique trophy. She might keep him in a magically bound enclosure and taunt him constantly. Harry, in this situation, would have to rely on his wits and any hidden magic he might possess to try and escape. Maybe there are other Death Eaters who are jealous of Bellatrix's new 'pet' and try to harm Harry. This could lead to a series of events where Harry has to navigate the dangerous world of the Death Eaters while being in this subservient position, all the while trying to find a way to turn the tables on Bellatrix.
One possible plotline could be that Bellatrix captures Harry during one of his adventures and decides to keep him as a pet for her own amusement. She might use dark magic to bind him and make him obey her commands. Another could be that in an alternate reality, Harry is somehow magically enslaved to Bellatrix from the start, and the story follows his attempts to break free from her control while she uses him for various dark purposes, like spying on the Order of the Phoenix.
A possible plotline could start with Bellatrix having a strange fascination with Harry from the start. She manages to get him in her clutches and treats him like a pet. Harry, on the other hand, uses his wits and the limited magic he can access in this state to outsmart Bellatrix. There could be a lot of psychological battles between them, with Harry trying to understand Bellatrix's twisted mind while she tries to break his spirit. And all this could be set against the backdrop of the ongoing war between the good and the evil in the wizarding world.
One scenario could be that a powerful magic spell backfires and turns Bellatrix into a small animal that Harry then takes in as a pet. Another could be that in an alternate universe created in the fanfiction, the power dynamics are so different that Bellatrix is somehow subjugated to Harry and made into his pet.
Well, it depends on the specific fanfiction. There are countless fanfictions out there with all kinds of strange and wonderful ideas. Some might indeed depict Bellatrix as Harry's pet. It could be a part of a plot where magic or some other factor has changed their relationship completely. For example, a spell gone wrong or a new magic system introduced in the fanfic could lead to such an odd relationship.
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.
One possible plotline could be that Harry discovers the truth slowly. At first, he might notice strange similarities between himself and Bellatrix or some unexplained magical abilities that seem to link him to her. This discovery could throw him into an identity crisis as he has always thought of himself as an orphan with no real family other than the Dursleys.
In some fanfictions, Bellatrix might save Harry because she is magically bound to him in some way. Maybe a spell backfired and now she has to protect him. Or perhaps she has a secret love for Sirius Black (Harry's godfather) and in order to honor his memory, she saves Harry. There could also be a plotline where Bellatrix is tired of Voldemort's tyranny and sees Harry as the one who can overthrow him, so she saves him for the greater good of the wizarding community.
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.
Perhaps Snape, in a moment of weakness or out of some deep - seated sense of responsibility, decides to protect Harry in an extreme way. He treats Harry like a pet in that he provides for all of Harry's needs but also restricts his freedom. This could lead to a lot of internal conflict for Harry, who is used to being independent. Meanwhile, Snape has to deal with the other characters in the wizarding world who might not understand his motives for treating Harry this way.
A possible plotline is that Bellatrix is actually an undercover agent for the Order of the Phoenix all along, and her relationship with Harry is part of her cover. They get married as part of a larger plan to infiltrate the Death Eaters more deeply. As they live together, they start to truly care for each other. Another plotline could be that a powerful magical artifact forces them to marry. They initially hate each other but as they are forced to co - exist, they discover hidden sides of each other. Harry might see the broken and conflicted person inside Bellatrix, and Bellatrix might see the true hero in Harry. This could lead to a very complex and emotional story of love and redemption.
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.