How to find/change the apple id used to install an app?

Open the App Store on your wife’s iPad. Tap her Apple ID symbol (initials, photo …) in the upper right. Tap “Purchased” followed by “My Purchases”. Those are the apps she is entitled to have. Other apps should be deleted and, if desired, re-purchased (possibly at zero cost) by your wife.

Or perhaps this solution would work better in your situation.

How to share apps and purchases with your family - Apple Support

LesMikesell wrote:

The apple id 'symbol' shows her initials, but when I touch it to show the details, it is my apple ID - and it may have been that way for years.

This sounds like an issue that needs resolution. It would also be worth checking what happens when you perform the same action on your Apple device.

It certainly sounds like at some point there were two people using the same Apple ID, and that mistake was later corrected - but not completely.

Here’s what I recommend:

1. Continue the process I mentioned above. This will show which apps are hers. Note which apps she has and are not hers, as this list will be needed later.
2. Be certain all the material she needs is backed up.
3. Thoroughly cleanse her iPad as follows: What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support
4. Set up her iPad as a new device, using her Apple ID.
5. Restore the iPad backup.
6. Delete any apps that she did not own.
7. Re-“purchase” (at zero cost) those apps from the App Store.
8. Be certain your Family Apple One subscription is still intact.