On the Topic of primates there is Koko. She is an ape that has a pet cat. This is essentially a real life Mickey Pluto relationship.
Better yet, there are baboons who steal the puppies of feral dogs and raise them as pets, with no humans involved! Really. The video is less cute than you may expect, as the baboons are not great dog owners; they pull the puppies around by their tails as they yelp in pain.
Chick-Fil-A actually averts this by doing a series of ads with sentient cows shilling the restaurant... because they want people to eat more chicken and less beef.
Then there's the infamous "Miss Piggy advertising a bacon breakfast at Denny's". Watch it here.
There is a subset of BDSM called "pet play", with varying degrees of "sentience" alloted to the submissive. Some are humans who happen to be owned by other humans, others act and sometimes disguise as animals like dogs, cats, horses and cows. Maybe Pluto is just a sexual deviant... and so is Mickey.
This is one of the core debates that splits the Furry Fandom; as the definition for "anthropomorphic" in the community is fairly strictly textbook, arguments are over how far to one end or the other of the human/animal spectrum one has to slide before the character stops being considered anthropomorphic. Currently, the extremes are loosely set at Talking Animals at the extreme animal side, and just short of Cat Girl for the extreme human side (for some reason, pure Cat Girls are not considered anthropomorphic for most. Must be the human face that ruins it...).
There was one furry story involving the C-snakes being hunted down by The Man. The protagonist is the only host capable of thinking straight, and after comes up with the bright idea to infect one of his family's non-anthropomorphic dogs and send it off to divert the hunt. If the ridiculous idea of a furry buggering a dog wasn't ridiculous enough, the dog is later caught by dogcatchers and put in a pound, where it infects the other dogs, the anthro wolf dogcatcher, and they all go one to infest the city at large. It's exactly as ridiculous as it sounds.
Us. Well, more specifically, our relationship with non-human great apes. Despite humans and their ancestors being a member of this group, the others are less intelligent and are seen as "mere animals" even though they make and use tools, have cultural traditions, and (in the case of chimpanzees) fight "wars". This actually has significant real-world consequences, especially regarding the treatment of apes in captivity.
Neanderthals, during the period we coexisted with them, take it Up to 11. If they were, indeed a separate species from us, how did early Homo sapiens perceive them—as "animals" or as fellow humans?
We did interbreed with them, so if they were considered "animals" that would indicate some sort of bestiality. Of course, that doesn't remove the possibility at all.
Even more with the case of Flores Island's Homo floresiensis, who was the size of a child and had the brain capacity of a chimpanzee. It was believed once that they coexisted with modern humans in the island for a time, maybe even thousands of years, but the evidence has since been debunked. They existed in their island until 50,000 years ago, when modern humans had colonized Africa, most of Eurasia and Australia, which still makes it possible for them to have encountered modern humans.
During the Congo Wars of the 1990s and early 2000s, there were so many instances of cannibalism of pygmy peoples by non-pygmies that some pygmy collectives requested the United Nations to formally declare cannibalism a crime against humanity. There are more rural Central Africans than you'd think who believe that pygmies are just another jungle ape and therefore susceptible of becoming bushmeat.
There's a class of ants that domesticated caterpillars and use them as cattle. They take them out once a day for them to feed and then they get them back inside the anthill during the night and they "milk" the honey they produce.