GA instructor with student reported an NMAC while doing pattern work at SNA airport. Reportedly; ATC advised them of the traffic; but both the instructor and student identified the wrong aircraft as the traffic. When they identified the correct aircraft the instructor took over the controls and executed a 360 maneuver to the right to avoid a possible collision.

Narrative

Student and I were doing pattern work and landing practice; on one of our laps in the pattern; Tower called up to extend our 15 degree (SNA procedure) upwind and reported traffic that will be entering the pattern off of our left and to turn downwind when traffic is in sight. My student and I both thought we saw traffic off of our left; reported the traffic in sight and proceeded to turn downwind. When we rolled wings level on the downwind; Tower informed us 'do you still see traffic in sight'; and then I saw traffic ahead of us in close separation and realized that my student and I both were looking at the wrong traffic when we initially reported traffic in sight. Due to the close proximity; I took controls from my student and made a right 360 maneuver to increase separation between our two aircraft. We continued following that to a safe landing.In the future as a learning experience from this event; I'll be sure to scan the area traffic is reported for a far longer period of time; and even reconfirm with Tower if we are looking at the proper traffic to follow to avoid any similar events in the future.