Flight Instructor reported an NMAC during landing pattern training. The Instructor and Student mistakenly identified the incorrect traffic to follow which resulted in the flight crew executing an evasive maneuver until flight was resequenced for another approach.

Narrative

I was instructed by tower to enter the right downwind for Runway XX at ZZZ. Once done; I was told to look for an aircraft on the mid-field left downwind; and that that airplane was #2 to land. I saw what appeared to be an airplane crossing the threshold to land; which I considered to be #1 aircraft. I then saw an airplane in the downwind; which I considered to be #2 to land. My student agreed and I instructed him to advise that we had the airplane in sight. We was told by ATC to follow that airplane; and that we would be #3; cleared to land. My student and I turned base with no issue; I did a visual scan for traffic in front and to the sides before turning final. Upon turning right to join final approach; the high-wing on the left side of my aircraft raised to reveal an airplane already established on final approach. Seeing this; I took evasive action; continuing the turn to the right; configured the airplane for climb and cleared the area until vectored to land. I believe the airplane ahead of me was on an extended downwind to make room for departing traffic. It is possible that I mistook a landing aircraft with one which was actually departing. It is possible that the controller momentarily lost track of the aircraft positions. I was not told to extend downwind. It is my understanding that; due to radar limitations; the ZZZ tower requires position updates from aircraft in the pattern. This would exacerbate issues in situational awareness. Being told I was #3 to land; I saw what I believed to be the #1 and #2 traffic; which; looking back; I should have double and triple checked. Had I paid closer attention to the locations of all aircraft landing and departing the airport to ensure maximum situational awareness; it would mitigate the effects of any possible mistakes made by others involved.