Flight Instructor reported making evasive maneuvers 'several' times to avoid another aircraft in the pattern. The Instructor reported the Tower Controller's instructions caused one of the NMAC's. The reporter stated Flight Schools' use of similar call signs contributed to the confusion.

Narrative

ZZZ was very busy the afternoon of the event; and had a single controller on both Ground and Tower frequency. We were doing touch and go's in the right hand pattern for Runway XX. After our third touchdown; we climbed normally for crosswind and downwind; suddenly seeing another aircraft turning downwind head-on from the opposite direction 100 ft. below us. He apparently had been instructed to enter the pattern from the east. We took evasive action and turned downwind inside this aircraft; while they did the same and turned downwind next to us. We had to take evasive action on downwind several times as we tried to communicate and avoid colliding with the other aircraft who was now paralleling us very closely. The other aircraft was also communicating the predicament; but because there were a lot of airplanes and it's tail number sounded like others; we were not sure which one they were. Tower told them to take a left 360 on downwind; which they did; and that sent them straight at us because of an easterly wind. We took further evasive action and communicated urgently we were in conflict. Tower instructed us to turn base; to follow another aircraft on final and to land; which we proceeded to do immediately. Unbeknownst to us; as we rolled onto a 1.5 mile final at 800 ft.; Tower had meanwhile instructed the conflicting aircraft to follow their original left 360 with a right 360. They were now only 900 ft. or so directly over the final approach path. We were blind to each other at this point; the conflicting plane in a right turn above us; and us in high wing Cessna rolling right onto final. We evaded them again as they filled the windscreen; communicated to Tower we had no idea what was going on; and he responded 'That aircraft is of no concern to you'.I am a 3000 hour CFI; and the other person flying with me was a 25;000 hour CFI. We knew at that point Tower had clearly lost control of the pattern. This is our home airport; and we are very familiar with this Controller who is highly experienced. We have never had this close a call in the pattern; after several thousand hours of instructing here. The other aircraft was repeatedly sent into our path in close proximity. Please also note: at this very busy airport; Flight Schools have been allowed to name many of their aircraft with nearly identical tail numbers. It may have been a cute idea by a non-pilot; and why the FAA allowed it is insane. As well; Tower is understaffed and has been for a long time. All of this contributes to an accident that is waiting to happen; and most of the pilots who fly here agree with that sentiment.