App for messaging without using phone number?

Messenger is chatting based on your FB, also Instagram you can DM people, and google you can chat via google chat through email :)

Hangouts is now more targeted to business users, but it still does the job if you want something that's platform-agnostic, and using your Gmail address to log in.

Great design, cross-platform, secure AND you don't have to share your phone number.

EnjoyUp is still in beta but there no need of a phone number, only an e-mail

Just an email to create an account and awesome features :) Still in beta.

A great and highly underrated messaging platform

VIPole works with an email address and they're currently working on making an annonymous entry. It's a cool app that protects your identity and all the information you share within is encrypted e2e. Plus they have useful features like wiping up history, editing, message auto-burn. Althought you dont need a phone number to register, you can still make calls with audio or video and make calls to other people who are not using the app. These communications are also fully encrypted even on server, which means no one can really track you.

You can use your mail id itself.

You can try this one, users can use emails to register accounts and they do messaging easily in the app.