If your device isn't rooted, you'll want to use the Helium app (free, but you'll have to get the resulting files off your device by going into your file manager of choice and sharing them to email or the cloud; it will back up to cloud storage if you buy the app). This is a little fiddly to deal with, unfortunately; among other things, it requires you to hook your device back up to your computer whenever you restart Android. But it does work for Zen Koi--I tested it by transferring my game data from my tablet to my phone successfully.
Note: any app data backup, as far as I can tell, only works for data associated with the primary user of the Android device. If the game is being played on a restricted account or even a regular account that isn't the one that was originally set up, you can't get the data off.
(I have no idea what the situation on iOS is.)