Google Play is asking me to buy the license again, even though I have already paid for it.

If the license was recently purchased on another device, take a look at this article for detailed troubleshooting steps.

Otherwise, here are some things to check:
  • If the purchase was recent, double check to make sure it was truly successful.  You can verify the purchase by logging into your Google Payments account (payments.google.com).
  • Your device may no longer be linked to the same Google account as it was when your purchase was made.  You can check in the Accounts area of your device's settings.
  • If you are using multiple Google accounts, see this article for additional troubleshooting.
  • You may have synchronization of app data turned off.  This can also be checked in the Accounts area of your device's settings.

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