With the new user management system, you don't need to login with Amazon. You can directly login to the board with your login id and password.
Here is how to do that -
Connect to the board, with your login id and password
If you do not remember your password, you can click on 'Forgot password', provide your email address, and a password reset link will be sent to you.
To change your credentials after logging in, you can go to Settings -> My Profile. Here you can change your login id and password. You can keep the same email as your Amazon ID, or change it if you wish to use another email to login to the board.