Really? I thought registration asked for the Product ID, as opposed to
the Product Key (aka installation code, license key, whatever). Your
Product ID is in the About window of any Office app--About Word, etc.
By the way, registration does *nothing* at all but let you sign up for a
newsletter. Doesn't save your license code, doesn't activate the
product, nothing. Totally optional.
Re which product key is installed on which machine--trial and error is
your only option, sorry. See the steps here if you need to reset a license:
http://www.entourage.mvps.org/version/license_version.html#reset
I find that when I re-install the same license key, the Product ID is
almost identical except for the last 5 digits. However, the same license
key used on different machines might produce Product IDs that are more
different.