ADE has a nasty anonymous/implicit ID that it generates if you do not give it a 'real' Adobe ID;
that is what they call 'authorised without ID'.
It then associates that ID with any DRM book downloaded, and uses it for reading the book.
Unlike a real Adobe ID, this implicit ID cannot be used over multiple devices,
so cannot be transferred to an eReader or applied to another computer.
If you authorize your PC with your real ID, the PC will no longer have the implicit ID,
and so that is why you won't be able to read the book even on the original PC.
It is not possible to change the ID associated with a DRM ebook once it is downloaded;
either from implicit to real ID, or between real IDs.
When you first use ADE with the implicit ID it gives you a very weak warning about this,
but not it is not worded nearly strongly enough for users to appreciately the implication of what they are doing.
It's only when you try to apply a real ID that ADE admits to its own stupid behaviour.
The only way I know out of this situation is to use a DRM stripper, before you re-register the first computer with your real ID.
The legal status of DRM stripping varies from country to country, and in unclear in most (including US and UK).
It is probably technically against ts&cs associated with the eBook.
However, as long as you do not abuse it (for example by selling the book of giving copies to your friends),
but only use the stripped ebooks within the intended limit for which the DRM was applied, I think you are completely morally justified.
Adobe's implementation of DRM is so flawed that it prevents reasonable use of your ebook.
Once you have stripped this first ebook and checked you can read it,
you should then authorize all your computers and devices with your real ID, to stay strictly within the ts&cs for future books.