A TRACON Controller reported an aircraft cleared for a Visual Approach descended towards the wrong airport and flew below the Minimum Vectoring Altitude.

Narrative

I was training a CPC-IT on Area A; and the ZZZ area airports were southeast flow. We received Aircraft X from the northwest of ZZZ. We turned Aircraft X into the downwind northwestbound. The aircraft called the airport in sight below us over ZZZ1 airfield and in a turn from northeast to northwest. The trainee acknowledged this; but as we had been working on base and final turns; the trainee based Aircraft X about 2 miles north of ZZZ1 for the ZZZ visual approach. The trainee pointed out ZZZ airport at 3 oclock; 7 miles; and Aircraft X indicated it was in sight. The trainee issued a 100 heading to join final and cleared Aircraft X for the visual to XXR at ZZZ. A few moments later; I observed Aircraft X descending to 019 feet; which is abnormal for that distance from ZZZ. I then observed Aircraft X line up for XY at ZZZ1. I immediately cancelled the visual approach clearance and climbed Aircraft X to 020; pointed out ZZZ airport; and issued a low altitude alert. I also called ZZZ1 and informed them Aircraft X was correcting for ZZZ. Aircraft X subsequently got ZZZ in sight and was declared for the visual approach; landing at ZZZ without further incident. Looking back; we realized Aircraft Xs initial call of airport in sight was of ZZZ1 airfield. Even though the trainee did a great job of turning a base and turn to final at ZZZ and reissued the position of ZZZ; 6 miles further than ZZZ1; the pilots expectation bias clearly played a role in their decision to make a hard turn southeastbound and descend for ZZZ1 (a 2-3 mile final). Usually; a base turn almost over ZZZ1 alleviates this confusion but in this case it did not. In the future; I plan to place even further emphasis on pointing out both airports to every aircraft to further alleviate confusion. We discussed this in our training debrief and will continue to be vigilant in alleviating confusion and observing and correcting potential wrong surface events.