I think the concept is inherently not very privacy friendly.
I mean think about it, basically your AirTag sends some sort of identifier to all nearby devices and also sends along some sort of information how to contact you, the owner, either one the device can see directly or it has to send all of them to a central location and that location needs to know both the location of your tag and your identity to forward that information to you.
Basically, I want two things: to locate my cat wandering around town (more out of curiosity than anything else) and to locate my bike if it's stolen. So it has to be something small, or for the cat's collar, or something that fits in the bike's frame.
Maybe I'm asking too much, but it would be perfect to be able to link the device with the PhoneTrack app on Nextcloud.
Research dog/cat GPS trackers. There are several out there to choose from.
For you bike (bicycle?), I suppose you could hide a pet tracker in a mounted water bottle or tool bag under the seat.
Edit: fixed spelling error
It relies on nearby phones reporting the location of the AirTag. Even if you had the hardware for an open source tag, you aren’t going to have the ubiquitous network of compatible phones like Apple does. iPhones are everywhere.