You are right. It is, indeed, a real sport. The category depends on the video game. They can be in several different sports categories, from invasive to intellectual. Some require either strong hand-eye coordination fast reaction time, well-thought strategies, or all of them. But one thing is clear: video games at the professional level do require a lot of physical and/or mental skill. Because of this, and the fact that it is played professionally, with several leagues reaching a prize pool of millions, it classifies as a sport.
However, most people do not see it that way, due to the fact that it involves video games, often making them think "that is not a sport".