Why is port-starboard terminology used in aviation?
It isn't (much).

Ships are the most similar thing to aircraft that existed before them, so it made sense that aviation took over many terms and conventions from seafaring (the pilot in command is called captain, the cockpit is the flight deck, distances and speeds are given in nautical miles and knots etc.)

However since there is no physical port in an aircraft to serve as reference (most aircraft have doors on both sides), everybody has to learn to give the sides relative to the direction of flight and then it does not make sense to teach them new words when they already know “left” and “right”. So “port” and “starboard” are only used where there is a naval tradition.

I know British military aviation (RAF) used it (in wartime stories and documents it is common, but according to Simon's comment it is not common any more) and the US Navy uses it with ships, but I don't know to what extent they apply it to aircraft. 

However, it remains popular in simulator games, presumably because it is catchy and unusual and using jargon makes people feel special.