If you do a search on this forum you would come up with numerous posts very similar to this one.
The bottom line? You're using your AirTag for things it is not designed to do.
- Tracking - Since the tag relies on Bluetooth, and the Apple Find My Network, it does not work well for tracking a moving object. Whether that is a vehicle, person, or pet. It was primarily designed to help you locate a missing static object, like your car keys.
- Tag Discovery - When the tag is outside the Bluetooth range of the iPhone that it is paired with, it will after a certain amount of time will start emitting a sound (aka, beeping). This would be annoying to pets, but also to people nearby. Also, iPhones participating in the Find My Network (they do by default), will get a notification that an AirTag (not theirs) is nearby.
- Tag Issues - Pets have a habit to ingest things that they shouldn't. Should your cat find a way to remove the tag from its collar, it could try to eat it. Trust me, it can happen.
... so, if used as Apple intended it to be used, it is a very inexpensive device that can help you locate missing or lost items.