We keep score because there has to be a winner and a loser. The winner has to be recognized and awarded for it because they put more effort. People just say that so that the people who lose don't put so much pressure on them for losing especially if it's not a big deal.
If nobody wins, there is nothing to compete for, and no reason to play the game. So there has to be a score. Winning is not everything. Otherwise it wouldn't matter if one played fair or cheated. But it's an important part of sports.