It’s pairing keys that need to be copied over.
https://wiki.archlinux.org/title/Bluetooth#Dual_boot_pairing
Bluetooth devices use a hardware identifier (MAC address) to first find each other, then they negotiate pairing keys in order to keep communications secure from eavesdroppers. But creating the pairing key is done by the OS, and it’s stored with the OS’s files. So an alternate OS doesn’t normally or automatically have the keys a different OS created.
If you dual-boot but don’t copy over the pairing keys, then when you switch OSes your headphones (or other bluetooth device) still finds your computer using its hardware MAC address, but the pairing key doesn’t match. It can’t start secure communication, so it asks you to reauthorize and replaces the previous pairing key. Switch back to the first OS, and this happens again, because a bluetooth device only allows one pairing key for a given host / MAC address.