Why are humans the only species to maintain relationships with other animals simply for pleasure? In the wild, animals may share waterholes, and grazing areas. Some animals have symbiotic relationships one example being the hippopotamus and the oxpecker bird.
The symbiotic relationships are based on needs; Hunger, shelter, safety.

Human beings, appear to be the only species that has relationships with other species for pleasure. Companion animals and horses for riding are two examples.
Of course, there is also the use of animals to meet the species needs; eg dogs for safety.

In this question I would like to focus on the relationship between human beings and other species as a source of pleasure (or comfort).

What is different within the human brain that allows humans to maintain relationships with other animals purely for pleasure, that is absent in other species?
According to Hal Herzog, humans are the only animals that keep pets. Other animals have also kept pets however it was not under natural settings. These other "pet" relationships were observed in zoos, controlled experiments. etc.

One the main reasons Herzog notes that humans are the only animals to keep pets is the idea of culture. Humans have evolved to form a culture with social rules and norms. They can gather social information from others by relying on certain cultural rules. So it is acceptable for humans to form pet relationships with other animals. Some cultures do not have a word for pet. However ultimately these cultures end up forming relationships with animals that are "pet-like." The reason for this is because of memes. That is, the idea of having a pet spreads across different cultures.