The Greek word translated as “enemy” in the NT doesn’t mean the same thing as our word “enemy” in English. For example, it doesn’t mean a competitive opponent or someone who works against our goals.The Greek word translated as “enemy" is Ἐχθρὸς (echthros), which means "the hated", “the hating”, "the hateful", "the hostile", and "the alienated." The sense is someone that hates or is hated rather than just an opponent or adversary.
Jesus’s usual prescription for “hating” is “loving”. For example, in Matthew 5:44 But I tell you, Love your enemies. This applies equally both to those who are “enemies” because we hate them and to those who are enemies because they hate us.
Old Testament verses about enemies are very different largely because the word translated as "enemies" (o'yeb) has much more a sense of "adversary". In the Psalms, David asks God to destroy his enemies frequently.
However, the word that Jesus used that actually means “adversary” or “opponent” is “satan”. This is another Hebrew word meaning "adversary" but it is translated in a very misleading way in the NT. Jesus doesn't use this word as a proper name as we use it today nor was it used that way in the Old Testament. The fact that “satam” usually appears to be a proper name in translation is an artifact of translation, attempting to justify a certain view of Christianity.
Looking at the Greek, it is clear that Jesus himself uses this term "satan" generally as a title for anyone or anything that blocks or opposes us in doing God’s will. More about that word in this article. This is also the way the source word (which is Hebrew) is used in the books of the Old Testament.