C172 flight instructor reported ATC failed to provide separation in traffic pattern resulted in evasive action and a NMAC.

Narrative

Was doing pattern work with a student at ZZZ airport. I was on one of my last laps in the pattern on a downwind for XXR and the Tower cleared us as number 1 for touch and go Runway XXR while also advising us of traffic to look out for that was approaching the road and the highway which was headed northbound. We then read back our clearance: 'cleared touch and go Runway XXR looking for traffic; with our tail number.' Tower told us then to turn base inside the road; which we read back and did. We proceeded to do our touch and go; and then climb out for our right traffic pattern we had been previously flying. An extended upwind or specified crosswind was not specified by the Tower. As we turned a normal crosswind; there was converging traffic on the right downwind about 100 feet higher and passing in front of us as we were rounding from right crosswind to right downwind. I then advised the Tower about traffic while I initiated a descent. Tower replied: 'are you behind that aircraft'. I replied; yes behind that aircraft; about half a mile behind and he is 100 ft. above us (with end number). Tower advised us to do a left 360 and report reestablished on downwind. I read back the instructions and also advised the Tower that the aircraft is climbing. Tower then said: '(our tail number) that the traffic was our traffic to follow; it appears you cut him out; just let me know if you lose your traffic to follow next time; thank you.' In which I replied; 'My apologies (our tail number); did not see him on ADS-B; thought we had him in sight; but lost him; we'll uh continue the 360 to reenter the downwind.' Then once established on downwind again; I notified the Tower; and then requested a full stop and we got cleared and landed. We had a too close for comfort miss with an aircraft. Tower cleared us for #1 touch and go XXR and did not advise us the traffic to follow. The only traffic avoidance call we got was during our decent to keep an eye out for a Cessna transitioning north bound. Also did not tell us to fly straight out; extend upwind; or listen for them to call crosswind. Thus; we turned a normal crosswind into converging traffic.