Other mistake: After Ethan crashes the Impala, he shoots at Keifer to make sure he's dead. Keifer's head and torso are on the hood of the car, but the shots break the back window on the driver's side. (01:13:05)
Other mistake: When the two main characters are fighting each other with a pair of scissors near the end of the film, it is obvious that they are using rubber scissors - what is supposed to be a solid metal object shakes side to side. (01:41:55 - 01:42:50)
Answer: Yes, he must do this to truly feel like the different person he is trying to be. Also, so the police won't suspect anything if his traits don't match his identity.