I don't really hear too much about them themselves saying they are athletes. As far as a sport, a traditional sport maybe it isn't. But this day in age when things like chess are considered "sports" and the spelling bee makes it on ESPN, this more than qualifies. I watch so many tournaments on the world level and listen to countless interviews and watch their livestreams and have never heard them say they are an athlete.
I don't think sports are limited to physical activities anymore. At least not in my opinion. I think if you have to train at something for hours and hours every day to compete against others qualifies it as a sport.
I don't consider a pro starcraft or league player as an athlete, but I do think it is a sport.