Google Play:
Check that your account is authorised on the device and that the email and password are correct. Add an account if necessary: Settings - Accounts - Add account.
- Open Google Play Games app, tap the three dots at the top right corner to open the app menu and select Settings. Make sure "Sign in to games automatically" setting is enabled.
Make sure you have the latest versions of Google Play Services and Google Play Games.
- Follow the official Google instructions: Fix Google Play Games Mobile App sign in problems.
Amazon:
Open amazon.com and go to "Account & Lists" - "Your Account". Check the username and password of your Amazon account. Make sure there are no typos and that you are using the correct credentials.