PA38 Instructor reported a NMAC when a C172 failed to follow traffic resulted in both aircraft taking evasive action.

Narrative

While I was on a 3 mile final to runway XX at ZZZ; an aircraft on left downwind (C172) was told to spot an aircraft to follow on 3 mile final (My PA38). The Cessna called out 'traffic in sight' and proceeded to turn base. I suspected the Cessna spotted another aircraft on short final and was turning base in front of me as I could not spot the Cessna myself. I did not spot the Cessna until the moment tower called to the Cessna to go around and step left as they turned to follow the wrong airplane. I then went around myself and stepped to the right as the Cessna was within what I estimate to be about 200 feet off my left side at the same altitude. I believe the error occurred as an incorrect identification by the Cessna of the aircraft on short final as the aircraft to follow on 3 mile final. These errors occur at un-towered airports I suspect more often than at towered airports due to pilot complacency of expected safety at towered fields. I do not believe ZZZ has radar and if it did; I believe the error would have been corrected earlier. Had the controller not called a go-around; the distance between the two airplanes would have been much closer. If there were radar services at ZZZ; this report would not be written.