the same exact reason you say at mom's or at Dinner's. Let me break it down for you. So you got sport(noun) and then you got a car (another noun) and by adding an "s" at the end we get sport(')s (and now it's a possessive objective which means it denotes that the car in question belongs somehow to sport). Both are right, because sport is an objective, too, in word-combinations like sport(s) coat, sport(s) car, due to an s at the end is tacitly implied. You can observe the same pattern in other word-combination like a patrol car, fire truck, car pool ect.

"sports jacket" would be more like actual athletic wear, e.g. the jacket in a basketball warm-up suit (athletic pants and jacket players wear before the game starts). A "sport jacket" refers to a blazer (usually only men's style is called a sport jacket) and I think this originated from polo players since this is the kind of jacket worn typically. It is also called a sport coat.