client_id field of the auth_register is used in the DelegateWallet field of the add_authorizer transaction, while client_key is used in the PublicKey field of the txn. They need to be not in correspondence with one another, since this will mean that the authorizer is trying to self-delegate (i.e. use the same wallet for operations and delegation), which is not allowed.
This field should be renamed to delegate_wallet_id to be clear that this will not correspond to the client_key.