when the car with immobilizer got stolen, they probably do not disable the immobilizer,
bcoz, once my cousin got a Passo, it was with immobilizer but came only with one key (push start remote was missing), so we decided to make a new remote for it.
the guy took off the immobilizer ECU off the car, connected a IC programmer with his laptop, took off some IC from the Immobilizer ECU, and reprogrammed it to sense a new remote. (probably deleted the old missing remote configuration, and sets the ECU to sense a new remote. Then installed back the Immobilizer ECU to the car again, and by some procedure of lock/unlock ignition of two three times (some sequence for adding new remote), and then the new remote is working fine.
It was all done within 10-15 minutes.