C182 pilot reported they had misidentified traffic instructed to follow resulted a NMAC event.

Narrative

We had just departed from Runway XX at ZZZ and were initially instructed to 'fly straight out.' We were following a Cessna 172 aircraft that had departed in front of us and also was flying 'straight out.' A Cessna 206 had departed from Runway XX and was beside us on the runway heading and at a lower altitude. As PF I was maintaining visual separation from both aircraft; and was not receiving instructions from ATC for speed; heading or altitude. The aircraft directly in front of us; that had also departed from Runway XX; was instructed to 'turn right and proceed northwest.' The aircraft began the turn; moving to our 1 o'clock to 2 o'clock position. ATC then stated to us; 'traffic 1 o'clock; 2100 feet moving northwest; do you have them in sight?' I stated I did; since I believed he was talking about the airplane that he had just cleared to proceed northwest. I look at our TCAS and observed that aircraft was +3; or 300 feet above our altitude. We were climbing through 1800 feet; which meant the preceding aircraft was at 2100 feet. Once I confirmed I had that aircraft in sight; we were cleared to 'pass behind that traffic and proceed west.' I then entered a right turn and flew approximately heading 270. Once on the heading; well clear of the traffic we had been following; we were reprimanded by ATC (tower) that we turned improperly and were to follow traffic that was now passing below us. I look at our TCAS and Foreflight and observed a target without a tail number passing directly below us; 300 feet below. We were now climbing through 2100 feet; meaning that aircraft was at 1800 feet (pattern altitude). Based on the aircraft's location and direction as it passed; I believe it was likely at our 1130 or 12 o'clock position prior to our turn. I cannot determine if ATC called out the wrong aircraft; or if the aircraft we were supposed to pass behind was descending from 2100 to 1800 feet. However; in my opinion; had we not turned west we would have been much closer to a collision with the second aircraft that appeared to be coming inbound for a landing. Also contributing to this incident was excessive workload on the controller; as there was numerous training aircraft departures and numerous arrivals inbound during this time period.