The difference may lie in how as mammals, living with a dog or especially a cat is more similar to living with another person. Intelligent birds require a lot of interaction which is good for reducing mental decline, but the relationship is different — most birds spend a substantial amount of time in a cage and generally need to be controlled more, whereas cats and dogs usually freely roam about the house with a few areas that are off-limits.
I would wonder about dog vs other animals.
With dog you have to walk outside even if you have a garden, a dog needs a daily walk a bit further around.
Cat doesn’t need a walk, well it requires play time but I would say walking around the block should do more for person health than playing around with cat in-house.