Itunes Store keeps asking for logins i don't know

If the apps have all downloaded you could check your account's purchase history to see if anything is on the phone that isn't on it : See your purchase history in the App Store or iTunes Store - Apple Support

I'm not aware of any easy way of checking what content is causing the popup - that's one reason why it's generally best to have just the one account and not let other people copy content onto a device. Is there any content already downloaded that you don't recognise as having bought yourself ?

Nothing is showing as still downloading or waiting to download in any of the store apps ? The items must have been on the old phone when the backup of it was taken - had anybody else used the phone, or could somebody have added content to your computer's iTunes library and synced it to it ? In a computer's iTunes you can check what account owns an item by selecting an item (app, song etc.) and doing 'get-info' (control-I on a PC, command-I on a Mac), but you can't do similar on a device - if you have all of your content in your computer's iTunes library you could try that.

You are being asked for ids probably because there was content from them in the backup that you've restored to.