General aviation pilot reported a NMAC on final approach. ATC adjusted the pilot's clearance to avoid a collision.

Narrative

I was doing pattern work in a Aircraft X at a Class D field. There were two runways in use; 17L and 17R. This day Tower was keeping touch-and-goes to the east; i.e. left traffic. There are several flight schools; so as usual there were others in the pattern; so I was given an extended downwind to accommodate traffic. About this time Tower switched controllers. The new controller instructed me to turn left base; #2 for 17L. An Aircraft Y was behind me on downwind and was told that I was on left base and to report me in sight. The Aircraft Y reported me in sight; and was told to follow me and that they were #3 for 17L. When I was on final about 300 ft. AGL; tower abruptly changed my clearance to 17R and said that traffic ahead and to the left cut in front of me. I glanced out my left window and saw the other aircraft at my 10 o'clock strikingly close; maybe 150 ft. away. The Aircraft Y then landed on 17L and I did a touch-and-go on 17R without further incident. Had I not changed runways at the last minute; this incident could have been catastrophic. Aircraft Y likely thought the #1 aircraft ahead of me was the one to follow instead of my plane. Thankfully the tower was quick to catch Aircraft Y's mistake and subsequently divert me to the other runway.