C-172 Flight Instructor with student on training flight reported NMAC with another aircraft on final approach.

Narrative

Upon joining right downwind Runway XX at Traffic Pattern Altitude (TPA); my student and I were cleared for the option as number 2 for Runway XX behind another Cessna 172 on what we heard was 0.5 mile final. We both looked continuously for the traffic to follow and could not see the traffic on final within 0.5 mile of the landing runway. We then turned base diagonal to the Runway XX numbers at approximately 600 feet; and by then my student and I both observed another Cessna 172 in climbout just above the runway; and assumed that was the number 1 traffic we were to have followed; and that they had already made a touch and go that we did not see. We therefore continued to turn right final for Runway XX on centerline; and I asked the tower to confirm that we were indeed number 1 for landing Runway XX by that time; and they acknowledged in the affirmative. A few moments thereafter; another Cessna 172 behind and above us asked the tower if they were number 1 for landing; to which the tower requested they go-around. Thereafter; we landed the plane without incident; as did the other Cessna 172 following their go-around.In retrospect; I believe the traffic behind us was indeed the previously called number 1 traffic for landing; but that they were actually 1.5 miles out on final instead of 0.5 miles; which therefore was the reason we had not seen them (since we were not looking for traffic that far out on final) and ultimately cut them off inadvertently on short approach. The ZZZ Control Tower evidently had not noticed the encroachment of the number 1 plane behind us and perhaps mistook us for the number 1 plane; despite the difference in tail numbers.While I am glad that I confirmed that I was ultimately number 1 before landing; it was a mistake to rely on the tower to provide separation in that moment; since they too obviously had lost track of the sequence between us and the other C172. In the future; if I have any doubt as to the sequence and traffic to follow on final approach; I will be sure to query to tower again for sequencing or to call my base; if I have not seen the traffic by that time.