Your licence
Your account is your licence.
DiskDuo is bought inside the app, with the Apple or Google account you sign in with. There is no key to keep, and nothing to lose.
Moving to a new Mac
Install DiskDuo on the new Mac and sign in with the same account. DiskDuo runs on one Mac at a time, so it offers to move there, and the old Mac stops the next time it checks in. You can also free a Mac first, in Settings, under License.
Hiding your email from Apple
If you chose to hide your email, DiskDuo only ever sees Apple's relay address. Sign in with the same Apple ID on every Mac and it finds your purchase.
Refunds
Polar handles the payment and any refund. Once a refund goes through, DiskDuo stops working after it next checks in. Nothing on your Mac is changed.