Private messaging with no phone number, by design
Your phone number is the single most linkable identifier you own, tied to your real name, your address, your billing information, and probably your other social accounts. The instant a messenger collects it, your supposedly anonymous account is welded to your offline identity. That's why private messaging with no phone number isn't a feature in NoChat. It's the foundation.
Signing up to NoChat requires no SIM card, no SMS verification code, and no carrier interaction whatsoever. Your identity is a cryptographic key pair that lives on your device. People reach you by username or by a key fingerprint, neither of which depend on any phone carrier knowing you exist.