Tower Controller reported a solo student pilot turned onto an inactive runway after landing without a clearance.

Narrative

I noticed Aircraft X a student solo was not flying at traffic pattern altitude correctly the aircraft was at 2;000 feet MSL in a 1;200 feet MSL traffic pattern. The aircraft was requesting to land. During the landing phase Aircraft X turned left onto an inactive intersecting runway; Runway XX. I did not instruct the aircraft to turn onto the intersecting runway. I did not instruct the aircraft to turn because it appeared that the student pilot was trying to control the aircraft after landing and on landing-roll I noticed the aircraft was going side to side. I let the pilot do what they needed to control the aircraft and expected the aircraft to regain control and exit the active runway; which the aircraft did without issue. The student pilot solo was a bit shaken up by this and I allowed a few moments for them to compose themselves before switching to ground control. The OS (operations supervisor) at the time requested a pilot deviation be issued to the student solo and refused to do it. The OS called the flight school to issue a brasher statement.An educational phone call to inform the flight schools in the local area about exiting instructions will impact the future of student's pilots more rather [than] a possible pilot deviation.