The Play Store asks me to pay for the app even though I bought it

Sometimes, the Play Store doesn't reload the purchase status of the app and asks you to pay for it again. Even though you'd never be charged twice for the same item, and the Play Store would eventually tell you that you've already purchased the app and let you have it for free when you press the purchase button, there are a few things you can do to force the Play Store to update the purchase status.

First, you can try rebooting your phone and re-launching the Play Store, which usually helps.

Second, you can try going to Settings → Apps, selecting Google Play Store, selecting Storage & Cache, and pressing both "Clear storage" and "Clear cache". This should force the Play Store to reload the up-to-date purchase status from the Internet.

In addition, please check this FAQ, which could also be a reason why the Play Store asks you to re-purchase the app:

I deleted the paid app by mistake and I'm asked to buy the app again