Yes. You can use DAVx⁵ with one purchase on as many devices as you want, as long as they use the same Google account (or an account from the same Google Play Family Library).
However, sometimes Google Play needs some time to synchronize your purchases across all your devices. This can take up to several hours.
Google doesn't allow double-payments for the same product anyway. Even if you see and hit the Purchase button again, you will not be able to buy it again when you have already purchased it. If a second payment would really go through, this payment would be refunded automatically by Google Play.
- Make sure you use the same Google Play account (or an account from your Google Play Family Library) on the devices.
- Navigate to Google Play → open the navigation drawer → Choose "My apps & games". Press the reload button to search for updates → Library → If it is not displayed in the list continue with 2.)
- You can also try to trigger a synchronization of purchases by clearing the Play Store cache: Android settings → Apps → Google Play Store → Storage → Clear cache.
Always reboot your device after 1. or 2. and try to install again. You can also try to change the network to WiFi, which may trigger the sync of your purchases.
Even if you write to our support, sadly we are not able to do anything to speed things up. You'll have to wait until Google Play has synchronized your purchases.