You still need a phone with a registered Signal on iOS or Android initially to activate the desktop version (sorry if that wasn't clear), but you can turn your phone off after.
Edit: It actually has the option to register without smartphone, but it's only enabled in the debug versions.
Speaking of edits, you should probably edit the misinformation out of your original post, while you have time. It's ranked pretty high up the thread and it's completely inaccurate.
The original poster was correct though. Just because you only need to do it once doesn't mean it isn't required. As a person without a smart phone I can't use it.
Nor would I since it still requires a phone number to use.
You can build it yourself from source. While it's true that the debug version uses different servers, the functionality is there and can probably be enabled in a production build with little modifications.
You can get a google voice number and just activate through that. On iOS you have to type a confirmation code in manually, so there is no need for a debug version there.
Edit: It actually has the option to register without smartphone, but it's only enabled in the debug versions.