Does Facebreaker choose units?

Yes, Facebreaker chooses (targets) units.

According to RiftJudge FAQ #4087 and Rule 355.7, although Facebreaker's text does not explicitly use the word "choose" or "target," any instruction that requires you to designate specific game objects to affect is considered targeting in the Riftbound rules system. Therefore, when you play Facebreaker, you are targeting one friendly unit and one enemy unit at the same battlefield.