How do I associate a device with my Apple ID?

Question

I have an iPad and an iPhone.  I sync them with my iMac desktop computer, and never with any other.  I always use the same Apple ID.  As far as I know, it's the only one I have.
Recently, I purchased an app using my iPad.  The next time I synced the iPad with the computer, I got an alert saying "The purchased item 'App' on the iPad
'Etchemendy's iPad' could not be transferred to your iTunes library because you are not authorized for it on this computer."  The alert goes on, in fine print, to say, "To authorize this computer for items purchased from the iTunes Store, choose Store > Authorize This Computer."
So I choose Store > Authorize This Computer, and I get a second alert saying, "This computer is already authorized."  A nifty little continuous loop that neatly precludes me from using the app on any device except the one I used to purchase it.
But it gets better.  In stumbling around trying to figure out what's going on here, I accessed "My Account" using iTunes on my desktop computer (logging in with the same Apple ID I used to purchse the troublesome app).  When I click "Manage Devices," I arrive at a box that says, "These computers and devices are associated with your Apple ID for Automatic Downloads or to download previously purchased music, apps, and books..."  It then gives me a "list" showing just one device -- my iPhone.  The iPad is not shown here.  Neither is the desktop computer iTunes just told me is already authorized.  There's a button that says "Remove" next to the iPhone.  And two bigger buttons.  One says "Cancel" and the other says "Done."  There is no "add a device" option.
So how to I associate a device with my Apple ID?  And, as a matter of fact, why aren't all the devices I registered using my Apple ID already associated with it?  I never had any of these problems till the big iOS system upgrade a few months ago.  Since which, my interactions with Apple have nearly all been nightmarish.  Apple obviously put the WAY wrong guy in charge of iOS development.
Oh.  And BTW, when I look at the drop-down menu for choosing the "product" I'm having a problem with, only about half the devices registered with my Apple ID appear there.  And I have no clue what kind of iPad I currently have.  I have the warranty right in front of me, and all it says is "iPad WiFi 64GB Black." No doubt if I try to solve this using Apple phone support, I'll have to pay $20 to straighten out Apple's problem for them.  Which is why I'm here.

Answer 1

My guess is you're using a long Apple ID and just omitted or mistyped a letter sometime, or something similar.
Anyway, the way to recover is this:
You cannot change ownership of the mistyped Apple ID to the correct one.  After all, the mistyped one purchased it.  However, you can authorize purchases from the mistyped Apple ID on your devices by doing Store > Authorize This Computer on iTunes on your computer, and enter the mistyped Apple ID and password.  Then sync again and you'll be all set.
When you change computer (e.g., sell your current one and buy a replacement), you'll have to repeat this process.