Yes it's false, but what they probably meant was that there's no record associating transactions to particular people. They definitely could have used clearer language, but the story wasn't completely off.
Actually I think if someone *did* need to point at a record of *their own* transaction, they could sign a declaration that a particular transaction was theirs using their private key of the account the transaction was involved in. For privacy it actually might make sense that you cannot prove someone elses' transaction was theirs.
-MarkM-