Why Do think a sports car is called 'sports car'?

Because some sports cars can be used in the sport of car racing, which itself is a sport

This car is mostly used for sports only so sports car is called sports car

Basically it is called a sports car because the car salemen have found that that term will sell cars to a certain niche of customer.

A sports car used to be an agile car with great handling for racing on  the roads in England.  To be successful a car had to be small and light.  These fun cars were brought to America after WWII.  Young vets enjoyed them.   By the 1960's American  fun cars had branched in 2 directions,  The smaller (by American standards) cars that looked  low and sleek (sporty) like the Mustang and the bulkier Muscle Cars that did not handle well but had bid engines.

Neither of these really fits the idea of a sports car, but they have been sold as such.  

It has 2 seats, for 2 people.

A sports car is defined as a low small usually 2-passenger automobile designed for quick response, easy maneuverability, and high-speed driving. They are not designed for comfort or luxury. 

It's an attempt to associate the car with sporting events.

"Sport" implies the purpose is for fun and leisure rather than business or work.

For the same reason a "sport"-jacket differs from a suit jacket in that there are so many styles and colours and materials its basically anything goes,whereas a suit jacket really has to be the same colour as your pants and is meant to worn buttoned up with a necktie and collared shirt and dress shoes ,not the kind of attire you would choose for going out and having fun. Nothing to do with sport at all really.

In the early days of the motor car, there were those designed to be driven slowly on the public road and those designed to be used for motor sport on private roads and racing circuits.  The fast ones were called sports cars.  100 years later, the name is still used for cars designed to be driven at speed.

Because they are not comfortable for regular drives.

Because it's for people who like sports.