Well, it depends on the specific fanfiction. Some fanfic authors like to play around with the relationships and concepts in the Harry Potter universe. So it's possible that in certain fanfictions, Petunia is Harry's pet. However, in the canon story, Petunia has a very different role as Harry's rather unfriendly aunt who begrudgingly takes him in. But fanfiction allows for all kinds of creative interpretations.
One great Petunia fanfiction could be one where she has a moment of self - realization. Maybe she discovers some of Lily's old magic things and starts to understand the wizarding world better. It could show her in a more sympathetic light, perhaps she even tries to protect Harry in her own way.
One possible reason for Petunia never marrying Vernon in fanfiction could be to explore an alternate universe where the Dursley family dynamic is completely different. Maybe the author wants to focus on Petunia's own personal growth and independence outside of a traditional marriage. It could also be a way to introduce new and interesting plotlines, such as Petunia pursuing a career or having different relationships with the wizarding world.
I seriously doubt there is a legitimate 'Harry Petunia Piss Drink Fanfic'. Fanfic communities generally focus on positive and creative interpretations of the original works. This idea goes against the norms of good taste and the general spirit of fanfiction, which is to expand on the story in a meaningful and respectful way.
One popular theme is the exploration of Petunia's jealousy towards Lily. In fanfiction, this can be delved into deeply, showing how it might have affected her relationship with Harry. For example, some stories might show Petunia struggling with her feelings of envy for Lily's magical abilities and how that spills over into her treatment of Harry.
One common theme is redemption. Often, Petunia is portrayed as having a chance to redeem herself for her past treatment of Harry. For example, she might finally show some love or understanding towards him.
Petunia and pulp fiction seem quite disparate at first glance. A petunia is a beautiful, delicate flower that represents nature's beauty. 'Pulp Fiction' is a cult classic film full of violence, dark humor, and complex characters. If we were to force a connection, perhaps in a surreal short story, a petunia could be a symbol of purity in a world as chaotic as the one depicted in 'Pulp Fiction'. The flower could be a contrast to the seedy underworld shown in the movie, or it could be a mysterious element that somehow ties into the convoluted plotlines of the film.
By focusing on their shared past traumas. For example, both have had experiences related to Lily's death in different ways. Harry lost his mother, and Petunia lost her sister. They could start to bond over this common pain and gradually move towards a more positive relationship.
Well, in a lot of Petunia and Harry fanfictions, an alteration can be how their pasts are connected. Some fanfics might explore if Petunia had some sort of latent magical ability that she hid, and this could change how she views Harry's magic. It could also lead to her teaching Harry things about magic that she learned in her own way. Regarding Harry, his view of the Muggle world through Petunia could be different. He might come to understand and appreciate the Muggle side of his family more deeply, which is a big alteration from the original story where he was often at odds with the Dursleys.