GA pilot on final approach reported an NMAC resulting in them taking evasive action to avoid traffic which had been instructed to follow them.

Narrative

I was returning to my home airport of ZZZ on an IFR flight from ZZZ1. About 15 mi. out I cancelled IFR; as it was a clear day and contacted ZZZ Tower. They instructed me to join right downwind for XX and report midfield. I did that and they told me to continue downwind. It was quite a busy pattern at the time. Tower pointed out an Aircraft Z that was on a left downwind and turning base and asked me if I had a visual. I confirmed that I did and was instructed to follow that Aircraft Z in for landing on XX.Tower was also talking to Aircraft Y; who was further out to the southwest. Tower pointed asked the Aircraft Y pilot to follow the Aircraft X (me) on final for landing.As I was completing my base to final turn; I noticed my ForeFlight on iPad ADS-B showing a target virtually on top of me and colored RED. I pulled back the shade on my bubble canopy and saw the Aircraft Y right on top of me at almost the same speed (he was maybe 10 kts. faster) about 400 to 500 ft. higher than me and descending. I immediately increased my rate of descent to ensure separation and called Tower to report that there was an Aircraft Y on top of me and descending into me. The Tower instructed me to make a Right 360 and rejoin Final. Tower also responded that they don't have Radar; so cannot determine relative location several miles out. The Aircraft Y pilot never responded and continued to land.I rejoined Final and landed and taxied to my hangar. Once I calmed my wife (passenger) and parked the plane in the hangar; I planned to drive over to the FBO and discuss what happened with the Aircraft Y pilot. However; before I could do that; I saw the Aircraft Y take off. I did grab a screen shot of the Aircraft Y departing on ForeFlight with tail number and can provide that.There are two issues that I would recommend follow-up on. How did the Aircraft Y pilot mis-identify or not identify my Aircraft X that he was instructed to follow? I would expect that a turbo-prop aircraft would have ADS-B; TAWS and potentially other traffic warning systems that must have been giving lots of proximity warnings. The temporary Tower at ZZZ does not have any radar capability. I am certain that the vast majority of pilots have no idea and are expecting that Tower is giving them separation services. I would like to see this Tower get real radar or at least let them have an iPad with ForeFlight and ADS-B as 'advisory' equipment to supplement their visual observation of traffic.