I mistakenly registered my wife's and my iPhone under the same account on the same computer, albeit with different passwords. How do I separate these phones?

Question

This mistake has confused the account somehow and actions done on one phone often have an effect on the other. I took these phones back to the dealer where they were purchased a few months ago, and they were able to correct some of the issues, but we still have problems. The latest is that my wife password would not work on her Apple ID account so she changed it successfully, but a few days later when I tried to update an app on my phone, and in trying to sign in to iTunes, it showed her name instead of mine, and of course, would not accept my password.
When we first bought these phones about a year ago, I registered them both inder the name "R & P's iPhone" and tried to assign different passwords for the 2 phones, but as I recall, this was not successful. We let it go at that, inasmuch as we had not explored the phone's features much beyond making simple phone calls. But as time went on, and we began installing apps and using other features, this conflict has showed up more and more often. Now I am told by the dealer that one cannot register 2 phones on the same computer, which sounds odd to me, but whatever, we are prepared to start over & register the phones on different computers, under different names.
I have an iTunes account, but is this where you make these changes? All I see on iTunes is related to music & movies.
Thanks kindly for any advice.
R & P

Answer 1

I access a personal iPhone and iPad, and a business iPhone with the same Apple ID.
The iPhone's passcode lock feature is completely separate from another iPhone or iOS device.
All apps, all paid and free apps include DRM protection which is tied to the Apple ID that was used to download the apps. If some apps were downloaded with one Apple ID and then a new Apple ID is created, in order to download an app update that was installed with the original Apple ID requires using that Apple ID and password.