Sorry if I wasn't clear enough - you need a phone for initial setup. After that, you can throw away the phone (or put it in airplane mode for a less permanent solution) and the Desktop client will keep working. In contrast, WhatsApp Web sends messages via your phone, so your phone needs to be on and connected to use it. Not so with Signal.

That said, registration independent of a phone number comes up a lot here. Signal is primarily a mobile messenger, so the assumption that its users have a smart phone running Android or iOS is a valid one imho. It would be nice if people without a smart phone could use it, too, but implementing this in a way that maintains Signal's ease of use for all participants isn't exactly trivial (contact discovery is a big one). I'd appreciate it, but I can also understand why it's not #1 on OWS' list of priorities.