"Designed exclusively for tracking objects, and not people or pets"
(emphasis mine)
That’s just so you don’t sue the over your lost dog
I think it’s also for practical reasons: your dog needs to be near a person with an iPhone. If the dog is in the middle of the woods it won’t show up. Generally most objects require a person to move them and so the chances of them being near an iPhone are much higher.
Or your dog eating the AirTag with the button battery inside it