The ideal Mixpanel implementation that allows you to track users from anonymous browsing all the way through signup and subsequent logins is to call mixpanel.alias() on signup and mixpanel.identify() on login. Read more about standard use case identity management best practices.
But if you’re implementing Mixpanel on a site with users who have already signed up, you do not need call mixpanel.alias() on those users.
The primary purpose of calling mixpanel.alias() is to connect a user’s anonymous events from before a signup event with his post-signup activity on your site or in your app (i.e., linking the user’s previous identifier to a new identifier).
Since there is no concept of a previous identifier or anonymous events for already registered users, you do not need to alias existing users. You can simply identify them with the Mixpanel identify method, or whatever value you choose to identify them with.