the phone number is still going to be required for making an account, you can just choose to not share it with others and give them your username instead.

Quote from the blog post:

Registration Fees

Signal incurs expenses when people download Signal and sign up for an account, or when they re-register on a new device. We use third-party services to send a registration code via SMS or voice call in order to verify that the person in possession of a given phone number actually intended to sign up for a Signal account. This is a critical step in helping to prevent spam accounts from signing up for the service and rendering it completely unusable—a non-trivial problem for any popular messaging app.

SMS verification is expensive.

Obviously, running the infrastructure to support the entire user base is also expensive. Decentralized protocols like Matrix sidestep this problem by allowing anyone to host their own infrastructure to use the network. Even if the largest Matrix server shuts down, the network will live on, and people can migrate to another server or host their own. This distributes the costs and allows for different business models to support those costs – commercial, non-profit, cooperative, whatever. Corporations can (and do) host their own Matrix servers for their employees, for instance. I wouldn’t be surprised to see universities do the same, like they frequently do with email.

There’s an IETF internet standard for federated messaging called XMPP. Just be compatible with the standard. It also allows for extensions if you offer more than the core spec.

There’s a few forks that have done it. You could also look to Matrix to see how they’ve done it.