Hi Maggy, I think you've got it right (and ala_xfire is also sort of right). The SKREEM module (or immobilizer) looks at a serial number coded into the ECU, and if it matches the one it has stored, it lets the ECU start the car. That is the immobilizer and the ECU are coded together, unless they match then the ECU will not start the car. All the other stuff about the key fob and SKREEM is correct.
I've included a screenshot of an ECU that doesn't match the immobilizer. As you can see the immobilizer is preventing the ECU from starting the car because the Serial Number (displayed) stored in the ECU does not match.
DAS Drive Authorization