I'm not sure I follow, though I agree with your definition of idempotency I think ensuring idempotency on the receiving side is sometimes impossible without recognizing that you are receiving an already processed message, in other words: you recognize that you have already received the incoming message and don't process it, in other words: you can tell that this message is the same as an earlier one, in other words: the identity of the message corresponds to the identity of an earlier message.
Its true that idempotency can sometimes be achieved without explicitly having message identity, but in cases it cannot, a key is usually provided to solve this problem. This key indeed encodes the identity of the message, but is usually called an "idempotency key" to signal its use.
The system then becomes idempotent not by having repeated executions result in identical state on some deeper level, but by detecting and avoiding repeated executions on the surface of the system.