Purchased an iPad app compatible with Mac. it shows unpaid on macOS: I cannot redownload it.

I purchased StaffPad (https://apps.apple.com/fr/app/staffpad/id1442074103?l=en-GB) on my iPad. This is the iPad's App Store link.

I sent the very same link to my Mac, it takes me to the same app's page but instead of the little cloud to re-download, the price is shown.

All is done towards that goal: both devices on the same account, same profile picture. I also manually search the Mac App Store to search for StaffPad and filter by device: iPadOS and same result. On my other Mac, same story.

any ideas?