I don’t know about elephants, but it is telling that we apes can recognize ourselves in mirrors, while many others can’t. Is this a sign that one is beginning to learn to relate subjective experience to what it is that one sees?  If this is on track, then it is probably related to empathy, when one learns to relate something very similar between vision and being.  It passes for Theory of Mind in some circles, but I think that is a bit misleading.  It is self-recognition, and may be mistaken for self-awareness.

Look at yourself in a mirror.  Now try to imagine being a less social animal (for only social animals can do this), for whom that thing you see as a reflection has no significance.  The viewer does not try to make sense of that blob of light.  My immediate feeling is that this animal is not socially connected, and for the animal, he himself is not part of any picture he sees.  Compared to the non-social animal, we are a level higher.  We dinged.  We are dinging again, I think, as we recognize, out there, how we are formed.