Signal is really three things with the same name (four or five, if you count the foundation and/or the LLC):
1) The Signal protocol, which can have multiple implementations (WhatsApp, Facebook Messenger and Skype all use it),
2) The Signal client and server, which are open source, and
3) The Signal service operated by Signal Messenger, LLC.
You can use either 1 or 2 to implement a system that doesn't require a phone number to use. Wire, for instance, is a messenger that has adopted parts of the Signal protocol and the Signal source code. [https://github.com/wireapp/proteus/blob/develop/src/internal...]
What you can't do is change how the Signal servers operated by Signal Messenger are operated, or refer to your own client/server using Signal trademarks.
For all practical purposes, "Signal" being open source hasn't made that happen. As for Wire, yes, I know about its origins and I use it too, but even Wire had disagreements with Signal on using the protocol and finally managed to settle it (this is from 2016, so take it for what it's worth). [1] Wire is also focusing more on paying users and businesses and building features exclusively for that market. So the free tier is no longer a great place to be in.
