You can sign-in with Apple on your iOS device or if you own an android device, you can sign-in with your Google account.
This allows you to permanently save your game progress, and transfer it from one device to another, on the same platform (iOS or Android). Please note that it's not possible to transfer your game from an Android device to an Apple device or vice versa.
To link your account, go to the options menu and use the sign-in button next to your player ID. You'll see your account status update after the process is complete.
To transfer your progress to your new device, you need to sign-in first on your curent device, so your save is tied to your account. Then you can install the game on the new device, access the options menu and sign-in, with the same account that you used on the first device. The game will recognize the account and transfer your save.