Flight Instructor on final approach reported a NMAC with an aircraft on base leg for the same runway.

Narrative

This event happened on short final while doing traffic patterns with my student (pre solo); doing right traffic for runway XXR. On an extended right downwind for XXR; I received a base-turn approved; and an option clearance for XXR; and read it back. A Glassair aircraft then made a call that he was inbound to land. ZZZ tower then switched me to XXL; and the Glassair made an inaudible readback. Neither I nor my student read back the runway change call. A few minutes later we got a traffic advisory from ATC of a Maule on right base for XXR. Thinking we were still cleared for the option XXR; we read back that we had the traffic in sight; mistaking an aircraft on left base for XXL as the aircraft we were notified about. Between a half mile and a mile of the runway; my student pointed out an aircraft to our right. I immediately took the controls and performed a go-around. In the moment of seeing the aircraft; it was slightly below and starting its turn from base to final for XXR. The closing rate between the conflicting aircraft and I was very quick; and it looked to be within 2-3 fuselage lengths away. given the size of the aircraft within the windshield of my aircraft; I'd say it was within 100 feet horizontally and about 50 feet below; just at the 2 o'clock position of my aircraft. After performing a go-around; ZZZ Tower switched me to runway XXL and I terminated the flight. I reached out to ZZZ tower after the flight concluded; and also reviewed the audio transcript from the flight. My understanding is that the controller was being trained and under the supervision of another controller. This being said; I think that this is an example of many small things occurring in just the right way to lead to a potentially dangerous situation. The factors I believe had the biggest impact on this event were: I missed the switch to XXL; I believe the Glassair read back the landing clearance; although it wasn't very legible from the transcript; ATC didn't ask for a clear readback of the option clearance; and we confused the traffic advisory landing on the right runway for an aircraft landing on the left runway.