How to verify authenticity of AirPods 3 without a charging case?

So I lost the case of my Airpods 3. Since they were a gift, I wanted to check their authenticity before buying a new case. Unfortunately they have no charging left, so I had to resort to checking the authenticity via serial number. Firstly, I noticed the serial number in each bud is different. Secondly, apparently none of these numbers match the serial number on the case. My first question is, if that is right. I cant check the case’s number since it’s lost. My second question is, if it is normal that check.coverage does not recognize either of the serial numbers on each Airpod.

Thanks.

[Re-Titled by Moderator]