Based on the game by Black Lilith.

Do animals have pets like humans have pets?

Koko the gorilla is a sign-language speaking ape, believed by her handlers to know more than 1,000 signs- she kept cats.
According to them, she asked for one, chose it from a litter, named it, mourned it when it got killed by a car, then acquired 2 more kittens .
She took care of them, handled them very carefully.
I think this qualifies as keeping a pet.
There are videos on-line you can watch about her.
Of course, I have read of several signing primates self-identified as people rather than with their own species. Tonda, an orangutan that lived at ZooWorld in Florida, took in a stray cat named T.K., over time coaxing it into becoming friends, and they remained so for many years.
Tonda was not taught signing. Tarra the elephant had her pet dog, Bella a stray that wandered in at the Elephant Sanctuary in Tennessee- she even stood vigil after dog was hurt for the entire long time it was healing.
There is a video on this story I think. Amy the deer had her pet dog, Ransom, a blind Golden Retriever at an animal rehabilitation center in Oklahoma, Wild Heart Ranch.
Dog was born blind, she groomed him and plays carefully with him.
There is a video for this also I think According to a report, a crow raised a stray kitten that probably couldn't have cared for itself, but the only assistance it could have received was from a crow that never left the kitten's side.
The crow was seen regularly feeding the cat with worms and other prey that it had collected, and the two animals would often play together.
The crow would protect its pet from dangers, even squawking so that the kitten wouldn't wander into the road.
There is definitely a video to support this story. A group of capuchin monkeys in Brazil have been witnessed adopting and caring for a baby marmoset, a totally different type of monkey.
This is more of an adoption I think than a pet, and many types of animals have adopted and raised animals of other species. However, many animals become pals with animals of other species. I used to have 3 horses and 2 cats.
Every evening the cats came to the back door and waited, while 2 of the horses stood at their gate.
I'd come out all 4 of them would romp their way down through the grassy field with me with the horses and cats chasing each other around and the horses being very careful of the cats.
1 horse and 1 cat were particularly good friends and at times the horse would carefully stretch down to gently nudge and huff over the cat.
I have heard many stories of cats, dogs, goats, etc.
becoming stable buddies. Owen, the 600 pound baby hippo became close friends with Mzee, a 160 year old giant tortoise in a Kenyan game preserve.
Not what I'd call a pet, rather a pal. In nature, there are many cross-species relationships that benefit both sides.
Not exactly pets, but akin to what people do with other species. An aggressive, venomous, large type of tarantula called Xenesthis immanis keeps a tiny, frog named Chiasmocleis ventrimaculata around to keep up the house-As the spider eats its prey, discarded remains begin to pile up.
This, along with the spider's appetizing eggs, attracts ants and other hungry pests, which the frog eats up before any damage can be done to the nest.
In exchange, the spider does not kill and eat the frog. Several types of ants keep 'cows' in the forms of aphids, leafhoppers, etc.
even moving them around, protecting them, or feeding them so the ants can harvest sweet nectar the other insects excrete. Potter wasps have actually evolved pockets in their bodies, called acarinaria, specifically designed to carry new mites to their nest to protect the baby wasps.
Once deposited into a wasp's nest, the mites hang around, share the babies' food, and stand guard.
When an intruder wasp shows up to attack the babies, which they eat, the mites respond by swarming and biting the enemy.