Conscious of, aware of... isn't it the same thing? perhaps the term is self aware?
If you asked a thousand people what there earliest recollection was some would say birth, some would say something like playing on the beach when I was three some starting school etc so I think it has more to do with early memory.
I think babies are aware of as much as they can see, hear and feel but a. are incapable of articulating and expressing it due to a lack of speach and all aspects of fine motor movement and b the brain is insufficiantly developed enough to always remember the 'mundane' day to day stuff when we have grown up.