Can you authenticate the user in Salesforce communities using just the the email or a phone number? Yes, this is possible using the login discovery page. Go to Settings -> All Sites -> Workspace (in your site) -> Select Administration -> Login and Registration. Set the login page type as “Login Discovery Page”
Login Page Type Setup
Allow the system to create a default login handler and set-up one of the System Administrator as handler executor. After you do the above setting change, on accessing the portal, you would be prompted with a simple UI asking you to enter the email or phone (no password prompted, but you have to set-up the password during the registration as a backup)
Login Discover Page Prompt
You give your email and submit, Salesforce sends you the verification code and on keying-in the verification code you would be automatically logged into Salesforce as the appropriate user connected to the email address.
For any advanced customization, modify the default handler to accomodate the changes needed. e.g. If you intend to use the email address in a different field in user profile table than in the standard Email field etc. Make a note that for using mobile based login, the system sends the verification code to your mobile number and you need the message credits purchased from Salesforce.
Without the above entitlement in place, you may not login with phone number. The developer license doesn’t provide the message credits.