Authentication by a custom field I added a field called "phonenumber" to the user profile fields which I want to use for login purposes, too.

Usually Craft allows an authentication by username or email-address. Is there a way to add another field for authentication, e.g. the custom field phonenumber? The goal is to expand Craft to accept logins via username and email-address as well as the phonenumber.
The short answer is no. Craft does not natively support this, and there's no way to implement it via a hook (or event).

A quick browse through the code shows that the UserIdentity::authenticate method is where most of the action takes place (called from the UserSessionService::login method), and that's very much tied to using either a username or an email address.

Your only option is to implement a custom plugin, with its own controller endpoints, which handles login requests. You would then submit the login form to your plugin, rather than to Craft.

Such a plugin will, of course, need to follow all of the same security best practises as the current Craft "login" implementation, and be actively maintained in case any vulnerabilities come to light.