Can the Immobilizer be repaired?

Anything can be repaired; what problem are you having? Security light flashing? If so, I would disconnect the battery overnight and then reconnect it. This should reset the system. If not, you can try a hard reboot. Here is how:

https://www.2carpros.com/articles/step-by-step-guide-performing-a-hard-system-reboot-for-your-vehicles-electronics-systems

Please go over this guide and get back to us.

Thanks, I think I'll do it!
The question of whether the computer is a Bosch! Because my mechanic claims that nothing can be done with the immobilizer and there is no way to turn off the red key on the dashboard!?

That is incorrect. There is always a way to fix the car, that's why I like them. We should run a CAN scan. You can get a CAN scanner (Controller Area Network) which will work on most cars from Amazon.
Here is a video to show you how:

https://youtu.be/u-4syLc-ifQ

Please go over this guide and get back to us.