No, it's not a popular fanfic genre. The typical fanfic themes in the Harry Potter fandom revolve around the adventures of the main characters, their relationships with each other, and the exploration of the magical world. The concept of Harry being Snape's pet is quite out - of - the - ordinary and not in line with the general tone and themes that most fans are interested in. So, not many fanfic writers would choose to write in this genre.
I don't think it's a very common fanfic genre. In the Harry Potter universe, the relationship between Harry and Snape is complex, but the idea of Harry being Snape's 'pet' is rather strange. Most fanfic writers are more interested in exploring their antagonistic relationship or Snape's role in protecting Harry secretly, rather than this rather odd 'pet' concept.