So when does the receiver record the IDs? When it receives the message but before processing or after it's processed the message? If the former, then what if it goes down during processing? Then the other receivers will keep rejecting the message even though it's never processed. So now it's less than once. If the receiver records it after it's done processing, then it could go down after processing but before recording it in the DB. So now you have more than once.

Also, isn't the assumption here that you will have a reliable connection to a shared DB?

You can have engineered solutions that is pragmatically close to deliver exactly once but it's not "pure" -- there are still scenarios, however unlikely, that it will fail.