Just so we are both on the "same page." When you "add" an Apple AirTag, you essentially are registering/pairing it to an Apple ID. Once paired, the tag should show up in the Find My app on an iPhone, iPad, or Mac. The tag is associated to the ID, not the phone.
So, again the steps to add an AirTag was provided in the support article I gave you earlier.
You are saying that it "paired fine." That would tell me that you went through those steps and were able to tap on "Finish."
You should also have seen, that alternately, you can add a tag by going to the Find My app first. You should have to do both methods.
What I'm trying to understand is where the process failed for you.