I suppose the first order of business is to consider whether it is actually possible to be self aware, or as the online dictionary puts it: "an awareness of self with regard to position and time and place and personal relationships".
Pretty self obvious, one would think.
The first problem comes with our good friends the monists who opine "chemical reactions in the brain obey the first and second laws of thermodynamics. Self awareness is merely an illusion, and for it to exist, we would be getting something for nothing. That is impossible."
We shall come back to that one later.
The second group do not help us much either - the spiritualists. They opine that you are not your mind, but outside of it. By self realization you may attain a higher state of enlightenment, and be at one with the universe. They see self awareness as a sixth sense, which is in the process of evolving in humanity.
We shall come back to that one later, as well.
Perhaps our good friends the theists may shed some light? "Self-awareness is not sin, and it can be produced by nervous emotions or by suddenly being dropped into a totally new set of circumstances." Hmm. Perhaps we shall have better luck next time.
Being agnostics who just love the scientific method, let us look there next. The neuroscients are pretty confident that our personality lies in the Ventromedial Prefontal Cortex: but that is little better than than the monist position.
The sociologists take the simple view that self awareness is self evident, and spend their time analyzing how people cope with their own self awareness.
It is the biologists who give us some meat to work with. A fighting fish has absolutely zero self awareness, and a mirror in the fish tank can reduce a fish to a self imposed torn and maimed mess. Tigers have some self awareness - enough to regulate their behavior in changing circumstances. Chimpanzees have a great deal of self awareness, but not to such a degree as to be able to use reflexive verbs, And humans so much self awareness that they can use self reflexive verbs.
How do the biologists know? By observing behaviors, they devise laboratory experiments and then, using the scientific method, explain the behaviors they observe.
So why are reflexive verbs such a big deal? Well, we can teach animals a large number of words - up to 200 words to chimpanzees using American Sign Language. But even if you teach all the words that could be used to meet the syntax of a reflexive verb, not one chimpanzee has so far pulled off the trick. The hypothesis is that their self awareness is not developed enough to use reflexive verbs - a hypothesis that will be made false, if one should ever do it.
A reflexive verb is not difficult to humans. For example, "I bathe myself" causes us no problem. We understand that the concept of "me" is both the object and subject of a single verb. A construction that is common to every language on our Globe - except one: a primitive whistle and guttural sounds used by some African tribesmen.
As the minds of living creatures get more complex, they progress from no self recognition through to total self awareness. The added complexity adds abilities not seen in simpler ones.
And while man has the highest degree of self awareness, it is not a trait unique to mankind. Which sort of makes a mess of the spiritualist thought that we have some special, evolving, sixth sense that leads us to the path of enlightenment. Not that such a path does not exist, only that it is not human esp that gets us on to it.
And that our self awareness is sufficient for us to build and use reflexive verbs belies the monist notion that it is only an illusion. It might be illusory, but it creates measurable differences in performance in different critters. Once something can be consistently measured, it has taken on traits that make it no longer illusory.
So, we get back to where you started. Of course self awareness is a human attribute, for you are constantly aware of it. I just thought you might like science to confirm that your point of view is quite valid.