It is about connotation. I play Go against you. You are my opponent or, if it is a tournament, I could also call you my adversary(connotation is a bit stronger, less matter of fact like opponent, more around competition now). After the game, we sit down and have tea and laugh.

Now let's explore the connotation of enemy. Playing Go against you calling you my enemy is outright wrong. However, if you were my enemy, I would not drink tea and have a laugh with you afterwards. I wouldn't even be playing a game of Go against you, but rather trying to do other not so nice things to you.

Such is the nature of connotation. Being erudite helps here ;) .