If we assume this is a more family - friendly or just very oddly named story, perhaps the 'pet' could be a loyal companion to the male adult in the story. And the rest of the phrase might be some sort of misnomer or a very strange way of naming an aspect of their relationship, like the pet is always around the male adult in a very attached way, like a'sucker' in the sense of being very clingy.
Since the original phrase is so unclear and might have inappropriate connotations, let's reimagine it. The male adult could be a lonely person, and the pet comes into his life. The pet is so affectionate that it's always near the male adult, almost like it's'sucked' into his life. The story could then be about how this relationship changes the male adult, how he learns to care for the pet and how the pet brings joy and new experiences into his rather dull life before. This is all just speculation based on trying to make sense of a very strange phrase.
Let's consider that this is a story about a male adult and his pet. The 'pet' might play a crucial role in the male adult's life. Maybe the pet is always following the male adult everywhere, which could be what the'sucker' part implies in a non - vulgar sense. So the plot could revolve around the adventures or daily life of the male adult and his overly - attached pet.