WhatsApp always requires your phone to be around, whereas Signal needs it only when you link it. After linking, the desktop client is independent of the phone (being online or in your vicinity or the number being in your possession).
Yep, you're right. I just looked more into it and WhatsApp and Signal operate differently. WhatsApp works as I described, but Signal actually does share the original key between all devices through some sort of key sharing mechanism.
