Certainly a dog has more self-awareness than do insects.
A dog displays a sense of self during interactions with a dog pack. A hierarchy exists within a pack and every member knows its position. For this to occur, the dog must have an awareness of things existing outside and independent of itself. The dog must also conceptualize its life relative to the lives of other dogs.
A dog adjusts its own behavior in response to the behavior of another dog. The two dogs are working together. They are aware of their own existence and the existence of others.
The rules of the pack hierarchy are tacitly accepted, known, and enforced by all members of the pack. The dogs create something appearing quite similar to a little society-a system of interactions collectively conspiring to create a greater likelihood of survival and reproduction than pack members could expect were they to live alone.
Dogs seem to have selves because cooperation has been/is selected for in the milieus within which dogs exist/ed. A dog does not have the faculty or facility for language, nor does it have an opposable thumb. These are two things possessed by humans alone. These are two things making us unique. Yet still, the self-awareness of dogs seems of a different substance than your, mine and everyone else’s self-awareness.
I assert self-awareness is a condition required of mammals. I assert self-consciousness is a phenomenon separate from consciousness (i need to define consciousness here). I assert that humans are the sole possessors of consciousness but that this is and always has been subject to change. I proclaim self-awareness and its interactions with the reality in which it finds itself produced the conditions under which consciousness could arise. I say human consciousness is the evolutionary product of self-awareness experiencing reality through sensory organs. Further, it was these interactions occurring within the valleys and peaks of a very, particular fitness landscape and that these interactions, over time, created a milieu in which it was advantageous to have a more refined and intuitive awareness of self and the local environment. By advantageous, I mean, a more sophisticated self-awareness created more opportunities for survival until a reproductive age, more opportunities to reproduce at that age, and a better ability to ensure the survival and health of offspring until they grew.
I proclaim that self-awareness enabled society. But it was the complex system of interactions occurring between and among the social structures of a society and the humans whose interactions make up that society, which selected for the phenomenon which we call human consciousness.
Self-awareness led to society and society led to increased and more refined self-awareness. The further interactions of selves and society produced a complex system whose interactions resulted in the two novel epiphenomena we call “culture” and “consciousness.”