Nope, not a sport. Even though sometimes I think I am gonna have a heart attack from some
hands, it's not due to physical exertion. To me, "sports" have a strong element of physical skill. The physical exertion that comes from me leaping up from my chair and stomping around is not skillful. The manipulation of chips and skillful tossing of chips and cards is not sufficient physical skill/exertion to turn poker into a sport, as the challenge of poker is unrelated to the ability to do any physical acts.
Poker is a game. It is, or can be, an intense, stressful, enjoyable exercise in mental acuity, math, and applied psychology. It pits skill against the vagrancies of luck and the abilities of the other players. It's a pretty cool game. But not a sport.
Perhaps it was more of a sport in the days when most of the players carried guns and there was a chance you might have to run away with your winnings or become embroiled in a fight... or a shootout. The necessary skills back then included physical ones.