Yes, but you need to enable it first. When you start your game for the first time, your progress is not saved automatically to the cloud. You need to go to the Options and there to the Player tab.
There, tap the Google Play, Game Center, or Facebook icon to save your account. Alternatively, you can use your email address - enter it into the Create MADFINGER Account field. Once you link it, your progress is saved and you can recover your account (or use on another device) by logging in to any of the profiles you linked it to.
To get more info about how to restore your progress on a new device, check out this article: How to recover my lost account?