I don't think there is a special meaning for 向き合う in a sport or romantic context. Specifically, it doesn't mean "to connect". If you know a sentence where it seems to mean such a thing, please share that sentence so we can explain better.
向き合う just means "to face", and the object can be:
- another person, face-to-face (neutrally): 彼女に向き合って座る, 向き合って一礼する
- (figuratively) difficult situation: 現実に向き合う, 困難に向き合う
- another person with a difficult situation: 彼女に向き合う (this by itself can mean "to face her trouble" or "to face the trouble between her and me")
- (relatively uncommon) an inanimate but tangible object: 鏡に向き合って座る
In sport contexts, the person you're facing is usually your opponent, but that's not necessary. For example, a catcher and a pitcher are in 向き合う position in baseball.