Sport
1. an athletic activity requiring skill or physical prowess and often of a competitive nature.
Dance requires skill and physical prowess (to perform at a high level, but that's implied in the definition as well). In "competitions" it is of a competitive nature. Meets the definition.
One of the keys is that sport doesn't even have to be competitive. Duck hunting is considered a sport, but the only loser is the duck (or fish or pheasant or deer).
So, in dance, if you just work hard to get physically better at dancing, even though you don't compete, you are still taking part in sport.
It has nothing to do with whether it is "pretty" or not, or whether there is a winner or loser.