As far as I'm aware the protocol can use other identifiers. If I remember correctly the development server allows registering without a phone number for easier testing of Signal-Desktop. Not sure what is used as an identifier instead. Not sure where I read it, so it may be the result of the telephone game. Take this with a grain of salt.
Are you saying that the Signal Protocol created Open Whisper Systems, which is lead by Moxie Marlinspike, does not require a phone number to work? (Say the Signal Protocol instead of the browser extension since the browser extension uses the protocol to work.)