There is no method by which you can directly transfer an App between iPadOS/iOS devices.
When creating a backup of your device via iTunes or iCloud, the App itself is not backed-up per-se - instead, the backup contains a record of the installed backup. When restored, this backup-record causes the device to attempt to download the App from App Store servers; restoring the App is only possible if the App is both available and compatible with the target device.
Many “old” Apps have been (or are being) withdrawn from the App Store - either because the respective developer has decided to withdraw the App, or by Apple where the App is (a) not being adequately maintained or (b) fails to meet ongoing compatibility requirements. If the App that you wish to reinstall falls within these criteria, it cannot be downloaded or reinstalled.
Most, if not all, legacy 32-bit Apps are long gone; if not updated to support 64-bit devices, you are very unlikely to find the App within the App Store.
All App purchases are merely a “licence to use” for the duration of its availability.
If you still have a very old version of iTunes installed on your PC, from where you may have originally installed the App on your old device, it is possible that the App “.ipa” file may still be present. However, if the “.ipa” relates to a 32-bit App, it will not be compatible with a modern iPad or iPhone.