Will a second hand (but still genuine Apple, same model) airpods case work for my airpods?

Title says it all really, but for more information:

I lost my airpods case very recently (but I have the airpods themselves), for some reason FindMy isn't working ("Airpods Setup incomplete - some features aren't available).

Fortunately, I know to about a 20 minute window, when I lost it, and more or less exactly where I would have lost it. Unfortunately the location I lost it was public transport (I have submitted a lost and found request).

My Airpods are 2nd Gen Airpod Pros with the lightning charger socket.

On the apple website, it is saying a replacement case will cost £89. My question is, will a genuine Apple case, of the same model but second hand, still work, i.e. still pair as normal and still charge as normal (on the assumption it is in good condition). I ask just as I am not sure if cases become bonded to one pair of airpods and it becomes difficult to bond them to a different pair? Apologies - not so tech savvy.

Thanks!