Air tag not recognized by phone

When you put the battery in did you hear a chime on the AirTag? If not, then the battery is probably not making a good connection, which could be due to a coating like Rudegar suggested.

However, if you do hear a chime then there might be a different issue. Remove the back cover of the AirTag and pull the battery out, then quickly put it back in (you should hear the chime again). Repeat this five times and it should reset the AirTag. Then see if you can add it back to Find My.

some batteries comes with a bitter coating, so kids don't eat them, if yours do then this needs to be removed before you can use said battery in your airtag, if the coating is blocking the battery from working that would explain why it is not recognized by the iphone