Prop powered general aviation aircraft are far louder than automobiles. No pilot flies without a headset because the noise makes it otherwise impossible to communicate with the ground or other pilots. Peak levels exceed 100db and the continuous average is around 85db, which is damaging to hearing without protection. The above charts showing arrival and departure noise are for jet powered aircraft, not the light aircraft in the OP's question.
There are two sources of external noise, the prop and the engine exhaust.
Noise from the prop is caused by its tips approaching supersonic speed. Energy used to produce noise is wasted, so you'd think reducing speed would be helpful. Unfortunately, most piston powered aircraft have props fixed directly to their crankshafts for simplicity and reliability. These engines are already turning slowly (compared to car engines) when producing full power, so slowing them even more drops combustion efficiency faster than it is recovered by increased propulsive efficiency due to reduced noise.
Exhaust noise is caused in aircraft the same way it is in autos. However, the typical aircraft engine used in the trainers mentioned by the OP is a four cylinder turning more slowly, so the combustion events required to make high power are two or three times bigger. Your typical econobox car has a 2 liter motor while the aircraft motor here is almost 6 liters. Exhaust noise is consequently two to three times louder, unmuffled, and more difficult to silence because of the lower frequencies produced by the slow engine. Weight is expensive in an aircraft. Mufflers are heavy and reduce power from the even heavier engine, so the minimum acceptable solution is the one used.
Gearing a prop to allow it to turn slower and better mufflers could substantially reduce noise. These would both be heavy and raise cost not just for acquisition, but also in use as that weight has to be carried everywhere the aircraft goes. The pilot would still need to have hearing protection due to wind noise in the cockpit, which is not addressed as it was not part of the OP's question.