There is no "registration" as such. And there's no confirmation of purchases from the App Store outside of the password request when you try to buy it. So not sure what you're trying to do.
If you never changed the Apple Account that is being used for the App Store app to yours, then it's still using your wife's. You would have needed to erase the iPhone and set it up as your own. If your new iPhone is still using your wife's Apple Account for the App Store, then all the downloads are forever tied to it, and this cannot be changed.
For any new apps, you can if you really want to, change the Apple Account in Settings ➜ Account ➜ Media & Purchases and log in with your own, so purchases are tied to it, and any invoices are the sent to your account not hers.
Beyond that, there is no other assignment of property of the device or purchase confirmations. It may be that the account logged in to the iPhone still has her payment method added to it, and that is what is asking for confirmation for purchases, in which case you would need to remove that payment method and add your own.
click here ➜ If you need to change or update your Apple Account payment method - Apple Support
The other option, is that she is the Family Organizer in a Family sharing group and has Purchases sharing turned on and all purchases are charged to her. There is no way to change this while remaining part of the Family sharing group and purchase sharing turned on. You can add Gift Card credit balance to your account and it will use that before charging hers.
Of course this is all merely conjecture, as we have no idea what the setup is with your iPhone and hers, and whether or not you even setup a different Apple account when you started using her old iPhone or what kind of purchase confirmation you are receiving exactly.
If you explain exactly what messages you are getting or issues you are having with the purchase, someone may be an able to narrow down the issue and offer options.
