There are several definitions of "sport". One of them(from dictionary.com):
3. diversion; recreation; pleasant pastime.
certainly includes poker.
The primary definition:
1. an athletic activity requiring skill or physical prowess and often of a competitive nature
would certainly not include poker.
There are no sports based on that definition, however, that do not include some element of luck.
Poker is certainly a mental game. With all of the tools available for playing online poker, it has become more of a mechanical game that doesn't require that much thought. Much more mental effort is required to play live poker.