When Jesus said to love one's neighbor, he was asked who was one's neighbor. When he said to love one's enemy, he wasn't asked who was one's enemy. But while it may be unhelpful to work hard to identify people as our enemies, it is helpful to work hard to identify people who are not our enemies. After all, (a) it's hard to love those we take to be our enemies, and we shouldn't try to engage in this difficult task when it can be avoided simply by realizing that these people are not our enemies, and (b) love comes in a variety of forms, and the way one loves one's enemy differs from the way one loves a friendly or neutral person--and to love a friendly or neutral person as an enemy is to do them an injustice. So we should work to avoid incorrect identifications of enemies.
In particular, I think it's important to avoid identifying mere opponents as enemies. I am not quite sure how to define an opponent, but roughly x is y's opponent when x tries to prevent what she knows to be a goal (final or subsidiary) of y and, roughly, x is y's enemy when x tries to prevent y's flourishing. (And it's part of the concept of flourishing that it is a goal of that of which it is the flourishing.) I suppose that all enemies are opponents. But not all opponents are enemies. After all, there are multiple ways one might try to prevent what one knows to be a goal of y without intending to take away from y's flourishing.
An opponent can even be a friend. If I play chess against you, we are opponents: we each intend to keep the other from checkmating us while each knowing that the other intends to checkmate us. But that which constitutes the opposition itself can be a sign of friendship. We may play chess precisely because it is mutually enjoyable to one another, and it is only mutually enjoyable (barring deceit) when each is trying to win. I think this is the ideal case with sports and other games: each is extending to the other the opportunity of engaging in this worthwhile mutual enterprise precisely in opposing the other. Playing a game should, thus, be a kind of bid for friendship. (I understand that sports and games sometimes don't work that way--that's sad.)
Of course not all cases of non-enmity opposition are ones where the opposition constitutes a bid for friendship. One can have opponents who are neutral with respect to one's good: they pursue a cause and see one as pursuing an incompatible one and hence the oppose one's pursuit. But hopefully one can presume that the opponent only disagrees about a subsidiary end. And if so, that's a basis for friendship.
So perhaps we can say: Love your enemies, but don't mistake mere opponents for enemies. Strive for friendship with mere opponents.