is the serial number on the aripod pro earpiece and on the charging case is same

With AirPods (2nd generation) and AirPods Pro, each AirPod also has its own serial number, printed on the AirPod itself. If you need to send in your AirPods or charging case for service, refer to the primary serial number, printed on the case lid. You don't need the serial numbers on the AirPods.

Find the serial number of your AirPods - Apple Support

1. I apologize I read your question wrong. Go to Settings > Bluetooth. Find your AirPods in the list (for example, "John's AirPods"). Tap the Info button  next to your AirPods, and you'll see the serial number.

Each airpod will have it's own serial number so if something happens to just one then you will only need the one serial number. Ex. Your left airpod just quit working you would only need to report the one number. However if they both quit working then you could give both serial numbers or the primary one that is on the case. Sometimes it's only one airpod that is needing repair and by having it's own number you only need to worry about the one number.

Totally wrong.

Warranty coverage is under the serial number on the charging case.

The serial number on the individual Airpods is not used at all.