Do primates in the wild keep pets? I know some captive gorillas had cats.

Yes - I remember Koko naming her kitten as "all ball" (kinda like "fur ball"?!).

I'm not sure if a wild animal has ever been observed to have a pet, but I guess many intelligent animals value playfulness and companionship, and have evolved to have tolerance and empathy, so judging by Twitter, etc, it doesn't seem that rare to have inter-species friendships even between animals that might be considered as predator and potential prey.

Baboons have been observed to capture puppies from feral dogs and integrate them into their troop.

Interesting - I'd never head of this. There seems to be some debate over whether these puppies/dogs might be considered as pets, but perhaps it was the same for early humans too - domesticating cats (egyptians) and dogs for their utility at controlling pests and guarding/warning against intruders, then later warming to their furry friends.

Not just that, but there's a very wide, gray area in our other domestication stories too.

You don't just rock up to a herd of wild bovines and say "hey, you're my cows now, let me milk you". Rather, one day a group of humans started following a herd around, killing one once in a while for meat; eventually perhaps they started protecting the cattle from other predators and guiding where the herd went, while also starting to find ways to continue lactation in females and get them to acquiesce to being milked.

The cows that were compliant to the slow changes were the most likely to reproduce; rinse and repeat until you get whatever we have now.

It's been observed by one television documentary that, among other things, included a dog with a collar in one of these shots (with no explanation for the context).

There's no peer-reviewed scientific research describing this phenomena. Or even any attempt at gathering any data with something resembling scientific rigor.

A simpler explanation is that these dogs and baboons are just... Living side-by-side in a garbage dump. And that the film-makers just didn't include any footage that didn't fit their narrative.