The Donut Dog app is not designed to support multi-device perfectly, yet..
However, you just need to purchase it once.
I would not recommend to run the app at the same time on multiple devices! One device would override the other.
An example: You have 2 devices (A and B). You just bought PRO on device A but you do not see the PRO status on device B.
This is what you need to do to fix that: After you purchased PRO on device A, you need to trigger the sync process by running the machine (start, wait 10 seconds, stop, save the session). This will notify the server and upload the latest information.
After that, restart Donut Dog on device B (force quit the app and relaunch it).