Mr Autofire automatically saves your progress via Google Play Games (Android) or Game Center (on iOS). This means that your Mr Autofire account is tied to either your Google account or your Apple account.
To make sure you are signed into Google Play Games, open the Settings menu in the game. At the top you will see whether you are correctly signed in.
NOTE: On newer Android versions uninstalling and reinstalling the game does not remove the game files. Seeing your locally saved progress after reinstallation does not mean that you would be connected to Google Play Games, so remember to check the Settings menu!
To make sure you are signed into Game Center, open the device Settings application and choose Game Center from the list. The game also shows you a small "Welcome Back" / "Signed in as" banner on start-up.