Uniquely Unidentifiable
Aharon, if you follow the link to the Annotated IPG, you'll see a clear explanation of “uniquely identifiable”, and that the exception would not apply in this case - nor in your example.
As currently written, the IPG defines this as Drawing Extra Cards, and the downgrade does not apply (i.e., the identity of the card was not known to all players before being put in hand. This is a Game Loss.
d:^D