You're both right. The majority of bowlers, I think, if you really looked at it % wise would fall into a category in which bowling would be considered a game.
However, on the upper end where you get into competitive bowling, it is definitely a sport in that the physical conditioning and mental "toughness" come into play as much as in any other sport. Unfortunately, the % of bowlers who take the game seriously enough to elevate what they do to sport status is relatively small, and most recreational or non-bowlers will never see or understand that side of bowling.
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9~
It's a Sheep thing... You wouldn't understand!