Poker is a game involving luck and skill. Sports are games involving luck and skill. Example: Football requires a lot of skills to be played at a high level just like poker does. However, the luck comes into play in many situations such as tipped balls that get intercepted. Usually the ball isn't intentionally tipped to a defender but sometimes get intercepted by the mere chance that the ball would be tipped in such a way. Luck is what allows the worse team to win on occasion just as it allows a worse player to win in poker sometimes.
The lack of athletic necessity for poker is what primarily makes it lack the classification as a sport. Do you need mental endurance to do it for 10 hours in a day for a tournament? Yes. But this is found in many endeavors; many occupations and activities require mental endurance but fail to meet the standards to be classified as a sport.
Plus you could play a whole tournament and win without standing up more than a few times.
Not many sports where you can do that.