Doors – this door mechanism called

That's a Euro-style lock cylinder. It is normally held in place by a setscrew clamping it into the lock, but you may have to substantially disassemble the lock before you can get to that screw.

If you can't figure out how to remove the cylinder, I'd suggest dismounting the whole lock body from the door and bringing it to a locksmith. That will save you the "on site service" fees.

Rekeying double-sided Euro cylinders is a skill not all locksmiths in the US have. Single-sided, with keyway on one end and thumb turn on the other, are simpler.

Some modern inexpensive locks have an Allen screw holding the handle to the shaft. You would think that the Allen screw would be only on one side of the lock, the "inside", but in our locks it is on both sides, but the "inside" is what will give you access to the innards of the lock to release it. Loosen this Allen screw to release the handle or knob, then pry off the bezel to expose the innards of the lock.

On our Kwitset locks these Allen screws are loosened by turning CCW as you would expect, but the screws themselves are "left-handed". When the screws are turned CCW they screw into the shaft to clear the handle it and release it.