Where am I wrong in this symmetric circuit analysis?
How can we set V1 or V2 as short?


That's the rule when using superposition theorem.


  If I was asked this question I would first remove V1 and solve for the
  current through R3 and then remove V2 and solve for the current
  through R3 and add these currents. But that would give 5A which is not
  the case.


No, you have misused the superposition theorem. 

With V1 removed (and a short put in its place), the current from V2 flows into R2 + R1 || R3 and that is 5 volts / 1.5 ohms = 3.333 amps (split equally between R1 and R3).

If you do the same for V2 you get the same but for R2 and R3 and, this ultimately means that 3.333 amps flows through R3. But you would have probably seen that if you get rid of V2 and connected the right node of R2 directly to V1 - you can do this because the voltage sources (V1 and V2) are equal in value.


  How come a theoretical voltage source or a power supply in real pass
  DC?


That's pretty much what is normally expected of them. They do it with ease.