I am having the same issue as OP. I just got an iPhone X and am trying to restore a backup from a 6 Plus.
About a month or so ago, I edited my Apple ID to point to a different email address. The old email still exists, but I plan to close it soon and am slowly migrating services away from it. I also still have the 6 Plus from which the backup was taken.
When trying to restore my backup, I get a message that says "This iCloud backup includes purchases from the App Store that were made with a different Apple ID. Enter the Apple ID password for <insert old ID here> to restore these purchases." When I enter the password, it says "Verification Failed. Your Apple ID or password is incorrect." At this point, my options are "Forgot password?" or "Skip This Step". I have not selected the forgot password option because I haven't forgotten the password. If I select skip, it says "Are you sure? You must sign in to restore purchased content for this account." I have not proceeded beyond this, because I'm afraid of losing all my purchases, all of which were made prior to the ID change.
Here are some stone cold facts related to this. Please do not ask me if I am sure about any of this - I assure you, I am 100000% positive that all of the following are true:
1) I did NOT create a new Apple ID. I modified an existing Apple ID.
2) I am absolutely positive that I am entering the password correctly.
3) When I edited the ID, I didn't even change the password. So, the new ID and old ID have the same password.
4) The 6 Plus has not been asking me for any passwords for the old ID when updating apps, the way it would if they were purchased under a wholly separate ID.
5) I signed out of the old ID and into the new ID on the 6 Plus immediately after making the ID edit, and the iCloud backup I am trying to restore was created long after that.
6) Just to be sure, I deleted my existing backup from the 6 Plus and uploaded a completely fresh backup from the 6 Plus before posting this, to make sure that the entire backup was created after the ID change.
7) Once more for posterity, I did NOT create a new ID, it was an edit of an existing ID, and I am beyond positive that I am entering the correct password.
So... help? Thanks!