Can I pair my AirPods using a different charging case?

Hello,I lost my original case and bought one for 30 dollars. It works well to charge but not to pair. My left AirPod pro just stopped connecting out of the blue. I wanted to know if I could use the casing of my friends AirPods Pro 2 to pair my AirPods back to my phone ?

[Re-Titled by Moderator]

Original Title: Pair AirPods using different case