If you meet and talk to folks that are extremely passionate about their endeavors like:
running
sailing
swimming
biking
fishing
rock climbing
softball
rugby
etc....
they will all tell you their endeavor is an all consuming way of life that goes beyond the physical aspect and in to the realm of meeting a psychological need as well.
Which of those do you classify as a "sport"? They too, would tell you it is their "way of life". To some, they can take or leave running. To others, they "need" it, they study it, they are active on internet running forums, they read about it, their entire daily regimine is built around it. For others they feel the same for rock climbing etc...
Per the dictionary (thanks to Centaur on page 1):
Merriam/Webster says;
a physical activity (such as hunting, fishing, running, swimming, etc.) that is done for enjoyment
Certainly, bowhunting definitely meets that definition., our endeavor meets the definition of "sport".
Obviously, the word carries different connotations in it's application and there are many opinions on it.