App purchases are forever tied to the AppleID account from which they were purchased - and can only be (re)installed or updated when (a) you are signed-in to the Apple Store with the same AppleID credentials, or (b) if Apps have been shared with you via Family Sharing.
As described, due to authentication failure, you are currently neither signed-in to the Apple App Store with your Partner’s AppleID account credentials, nor have the Apps been shared with your own AppleID account.
AppleID account sharing is a bad idea at the best of times - as, even if you consciously know what you’re doing, you will encounter all kind of of complications and potential conflicts. This appears to be the situation that you have encountered. Be aware that in using the Forgotten Password link in relation to your Partner’s AppleID account, you will be attempting to reset your Partner’s account credentials - and risk locking-out both yours and your Partner’s AppleID accounts.
To unpick the mess, you might be best advised to fully reset your new iPad - and then again attempt to restore your existing iCloud backup:
Settings > General > Transfer or Reset iPad > Erase All Content & Settings
When restoring your backup, you will again encounter a prompt for your Partner’s account credentials; cancel the challenge and allow the restore process to complete. Clearly, items (such as installed Apps) associated with an AppleID that isn’t your own will not be reinstalled.
After the restore process concludes, check the AppleID credentials that are being used for the Apple App Store:
Settings > [Your Name / AppleID] > Media & Purchases
Immediately beneath the Media & Purchases menu item you will see the AppleID being used for the App Store. If this is not your own AppleID, tap the button and select Sign Out. After signing-out, restart your iPad:
Force restart iPad (models with Face ID or with Touch ID in the top button)
If your iPad doesn’t have the Home button, do the following:
- Press and quickly release the volume button nearest to the top button.
- Press and quickly release the volume button farthest from the top button.
- Press and hold the top button.
- When the Apple logo appears, release the top button.
Force restart iPad (models with the Home button)
Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.
After your iPad restarts, you can sign-in to the Apple Store with your own AppleID. Once signed-in, Apps that you purchased yourself will or can be reinstalled.
For Apps previously purchased by your Partner to be available to you, he/she will need to set-up Family Sharing - with you as a member of the Family Sharing group. Family sharing will allow selective sharing of purchased Apps, subscriptions and Apple services.
Family Sharing - Apple
Set up Family Sharing – Apple Support
Share App Store, iTunes Store and Apple Books purchases with Family Sharing – Apple Support
How to create a new Apple ID – Apple Support
https://www.imore.com/how-set-up-family-sharing-iphone-ipad
Once Family Sharing has been setup, you’ll be able to reinstall Apps on your iPad that were previously purchased from your Partner’s AppleID account - and difficulties that you have encountered this time around should not recur.
Summary
Sharing an AppleID account, if only for App Store purchases, is a bad idea - ultimately fraught with potential complications and frustration. While it is possible to use multiple AppleID accounts for different AppleID services, you would unwise to attempt doing so unless your personally own and control all the AppleID accounts that are being used.
