Changing push-pull transistors in brushless motor driver
I have heard many people working in repair shops that they prefer to
  change the whole set of transistors or to change some transistors even
  if they work well because they claim that even working transistors
  could have got deteriorated due to the burned transistor.


Yes, often the other FETs have been stressed and may be partially damaged. In particular, if one FET fails to a short (as they usually do) the FET(s) in the other leg of that phase (SW6 in your case) will have been driving into a dead short. Replacing both FETS - even when the other one appears to be OK - is a good policy.


  Should I change only the burned transistor or not?


Take the risk, or play it safe and do the lot? That's up to you to decide. 

When repairing a brushless controller I first examine all the FETS for physical signs of damage (burned or cracked case, stuff oozing out the ends etc.) and test each one for shorts or low resistance between Gate/Source/Drain. Then I remove the ones I suspect are faulty, put in just enough new FETs to get the controller working, and test it on a current-limited power supply with a low power motor. If it passes that test then I install the rest of the new FETs (if there were several in parallel) and test it again with full load.

In my experience, if more than one phase has a blown FET it is best to replace the lot. However sometimes even that doesn't work because the controller has some other issue, and then it is not worth the time and effort to repair.           


  Or to make sure that all the new transistors have the same
  characteristics so they will open and close simultaneously to get the
  right control signals for the motor.


Yes, unless you are sure that any differences are benign you should only use FETs with the same characteristics. In particular, Gate threshold, recommended drive voltage, and total Gate charge (QG) should be similar, to ensure that the new FETs turn on and off at the same speed. If several FETs are paralleled then they should also have similar on-resistance at the applied Gate drive voltage, to ensure that they share the current equally. 

Of course Gate/Source and Drain/Source voltage ratings must also be the same or higher. However higher voltage parts often have other differences that make them unsuitable.


  If I should change the other transistors as well, can I change only
  SW2 and SW4 since those are the ones that complete the circuit with
  SW5 and the motor coil?


No. If SW2 and SW4 are faulty then they will have stressed SW1 and SW2 as well. However the motor coil (unless also shorted) should provide some protection for FETs in other phases. It's worse for FETs in the same phase because there is nothing to prevent a massive 'shoot-through' current when one FET shorts.  


  Is there anything else that I should pay attention to so I can
  guarantee that the customer doesn't get back to me with another fault?


Make sure the controller can drive a full load with expected power and rpm for a prolonged period, with smooth speed control and no FETs getting hotter than others. A controller with weak FETs may appear to work OK at light load, but run rough or at lower rpm under full load.