Connecting two separate pairs of AirPods

I am afraid it is not as simple as that. Generally, the serial numbers of both AirPods and the case must all match for proper operation. That is why Apple needs to know the serial numbers when replacing one of them. From:

How to find your lost AirPods, AirPods Pro, or AirPods Max - Apple Support

"If you still can't find your AirPod or Charging Case, or if something is damaged, get the serial number and then contact Apple Support for a replacement. Then learn how to set up your replacement.

You can also learn more about AirPods pricing and service options."

See this:

Set up your replacement AirPods or charging case - Apple Support