Some jurisdictions require dogs/cats be "chipped" with an NFC tag along with a license and many more do it automatically with pets adopted at shelters. The tag, when scanned, has an ID that can be used to lookup the owner's info. It's usually implanted in the scruff at the back of the dog/cat's neck. It's not a tracker, but at least makes it easier to ID a lost animal with no tags.
If you are 3D Print inclined, our pup is outfitted with one of these on her collar:
https://www.printables.com/model/254459-airtag-case-for-backpack-camera-bag-strap-or-pet-c/comments
They say that to negate any liability for someone using AirTags to stalk another person, and/or someone trying to sue them if they did put it on a pet/person and the product failed to perform to their expectations. The warning is setting the expectation that they make no claim toward the product being reliable enough for that. Someone might place one on a family member with dementia or Alzheimers as a way to help locate them if they got lost. If there was no clear expectation that the product was not designed for this, Apple could be sued in the event the FindMy network failed and there was a tragic outcome.
If EM radiation was even remotely a concern, there would be warnings to not use AirTags in wallets or on keychains that go in pockets next to people's reproductive organs for hours a day. Not to mention iPhones and AirPods have UW radios too and have no such warning.