Just a short update after the engine replacement.
First of all, the engine was indeed replaced, I've got a letter from the dealer to use for rectifying the engine number with the authorities in the registration card, stating the new engine number. I've also seen the old engine with my engine number marked on it.
Did this help however? Well, maybe it did with the dangerously slow pickup which I did not experience after the engine change, however, I still have occasional engine hunting that would make the car "jump forward" in situations when the brakes are lightly pressed just enough to hold the car still at a traffic light and rough shifting, especially downshifts, especially in heavy traffic situations when it feels like aggressive engine breaking.
Additionally, the Brake System in conjunction with DST/DTC (4x4) lights came on in the dashboard a couple of days back. I took the car in for repair again on Sunday.
Warranty is over in a day so - I'm giving up!
I am considering a trade-in deal with a 2009 X5 offered by the dealer. It's not the best deal but I would say it's not that bad either - I just hope I can go through with it.
Thanks for those who tried to help and offered advice!
I hope I will have a better luck with another car.