Same issue here, reinstalling Machinarium from Play Store didn't help.
Logcat shows the following:
E/Volley (18845): [5774] a.a: Unexpected response code 403 for https://www.googleapis.com/games/v1/players/105963842337774641956?language=en_US
E/SignInIntentService(18845): Access Not Configured. The API (Google Play Game Services API) is not enabled for your project. Please use the Google Developers Console to update your configuration.
E/SignInIntentService(18845): com.google.android.gms.games.h.c.a
E/SignInIntentService(18845): at com.google.android.gms.games.h.b.a(SourceFile:153)
E/SignInIntentService(18845): at com.google.android.gms.games.a.bj.d(SourceFile:1682)
E/SignInIntentService(18845): at com.google.android.gms.games.a.bj.b(SourceFile:608)
E/SignInIntentService(18845): at com.google.android.gms.games.a.bj.b(SourceFile:585)
E/SignInIntentService(18845): at com.google.android.gms.games.a.t.d(SourceFile:822)
E/SignInIntentService(18845): at com.google.android.gms.games.service.w.b(SourceFile:349)
E/SignInIntentService(18845): at com.google.android.gms.games.service.s.a(SourceFile:47)
E/SignInIntentService(18845): at com.google.android.gms.games.service.GamesSignInIntentService.onHandleIntent(SourceFile:501)
E/SignInIntentService(18845): at android.app.IntentService$ServiceHandler.handleMessage(IntentService.java:65)
E/SignInIntentService(18845): at android.os.Handler.dispatchMessage(Handler.java:102)
E/SignInIntentService(18845): at android.os.Looper.loop(Looper.java:135)
E/SignInIntentService(18845): at android.os.HandlerThread.run(HandlerThread.java:61)
E/LoadSelfFragment(16402): Unable to sign in - application does not have a registered client ID
W/SignInActivity(16402): onSignInFailed()...
W/SignInActivity(16402): ==> Returning non-OK result: 10004
This is on Android 5.0.2, Google Play Services 6.5.99, Google Play Games 2.2.09.
If Amanita devs need help diagnosing the issue, feel free to contact me - I am a software developer, and I know how to use adb shell.