Yes, even if you un-install Monument Valley from your device and later feel like playing it again, you should be able to download the game from the store where you made the initial purchase without any extra charge.
You just have to make sure that you are using the same account you originally purchased the game with.
If you are logged into the right account and aren't shown a "download" button in the store but instead have a "purchase" button try logging out of your account, restarting your device and logging in again.
If that doesn't help, you'll unfortunately have to reach out to the stores support team so that they can check why you are being asked to purchase the game again.
We are unable to assist you further with this as we don't hold any billing information or have access to your store account for security reasons.