there were thin (~2mm) sticker-type Bluetooth tags [...] No battery, just a passive coil that could be found with Bluetooth signal and an app that shows how close it is.

Sounds kind of like UHF RFID, which are common in places like warehouses and can be done at a distance, but even higher frequency?

I imagine range was a huge issue. Unless you have an extremely powerful bluetooth tranceiver and a very high gain antenna (i.e. not a phone, a professional radio system), the inverse square law will mean you won't have enough energy to activate the electronics in the tag after a fairly short distance. Would probably work for finding something in your house though.

Sticker tag sounds like NFC. Certainly a bluetooth tag is possible too. Thing is, Bluetooth range is a few dozen meters tops. Airtag works by turning the entire Iphone population into a giant surveillance network, so if any iphone anywhere detects your airtag, it informs the Apple mothership, which in turn reports to you where your airtag was just seen.

Only comparable long distance approach I can think of without that is a cellular tracker. Those exist (search: cellular gps pet tracker) but it's a much bigger box with a subscription plan of its own. Someone else mentioned meshtastic so maybe that's an alternative in some places, or a LoRA tag instead of a bluetooth tag.

I think you must be confusing things or were told a lie. Bluetooth is a standard that is definitely not designed to be able to work like that. NFC is designed to work with thin tags that don't have a battery. Essentially they get power wirelessly from your phone in order to be able to send information. But, due to the inverse square law, you would need extreme amounts of power to be able to do that over a long range, so it just isn't something that could work with mobile phones.