[Prickle]

The case is usually just a case, or a charging device/battery for the ear buds.

The earbuds themselves are connected to your device via bluetooth.

Depends on the ear bud brand / cost. But they either both connect to your device separately, or only one of them connects to the device, then relays the information to the other bud.

[nosrepa]

Pairing is usually done through the case, hence OP's confusion.

[Prickle]

I googled the pixel buds and yea, that is a very weird and unique thing they have going on. It is definitely not the "usual" method here, as I have never seen anyone pairing via the case in my life.

I have never paired earbuds via the case either, but I guess there is a first time for everything.