Pilot on training flight in SR-20 aircraft reported NMAC with another aircraft in cruise.

Narrative

On a cross country training flight from ZZZ to ZZZ1; we had picked up flight following with Approach; and eventually handed off to [a different] Approach. We noticed a plane pop up on our Skywatch; 2 miles off our left wing; same altitude (5;500 MSL) and on a parallel course; no traffic advisory was given from approach at the time. Eventually we lost contact on Skywatch and then lost visual contact; until the aircraft popped up again on Skywatch but was now within 2 miles and appeared to be moving towards us to some degree. We regained visual contact with the aircraft just as we got a traffic alert from our Skywatch system; as the other aircraft was now on a perpendicular collision course and still at our altitude level. The pilot flying initiated an immediate descent; and the other aircraft never appeared to deviate course or altitude as they crossed directly over us at a vertical range of roughly 300 feet. No traffic alert was issued by Approach; and the Pilot Flying announced that we had descended to 5;000 MSL for collision avoidance. When queried about whether Approach had the other aircraft on radar they responded that they did not have until we announced our collision avoidance.