I assume when the OP asks about self-awareness, he is speaking of the kind of self-awareness that humans possess. He is not referring to the awareness of a slug that crawls out of the sun to avoid being dried up. St. Thomas assigns to all life three types of souls, the nutritive soul, the sensitive soul and the rational soul. While all animals have sensitive self-awareness, only a rational creature has rational self-awareness. Such awareness has (at least) a will, a conceptual faculty, and language. The will is what defines rational self-awareness when it conflicts with external forces and other wills. The ability to form rational concepts makes it possible to form a concept of self. The ability to speak and employ words has a two-fold purpose. Firstly, it is the building block of concepts. Some may argue that a concept is pre-lingual, however we only recognize them when they appear in the mind as words. Even if one disputes the necessity of language for the formulation of concepts, the second role for which language is indispensable is the communication of concepts. This is the major empirical test for conceptual ability. Whatever is purported to be conceptualizing must be able to respond to my queries, and it must be able to do so with a certain degree of loquacity. When Koko the gorilla says “give me candy” or Watson the computer says that the sun is 150 million kilometers from earth, I have grounds for doubting the rationality of these responses. However, if I ask something for an example of when it employed will, conceptualization, and language, and it not only understands the question but supplies sufficient detail to indicate a real experience, I would be more inclined to admit that it had rational self-awareness. We have yet to find a non-human that can do this.
The attempt to anthropomorphize animals is having an adverse and perhaps unexpected effect. The intention may have been to elevate animals in our understanding and thus have them treated more humanely. What is actually happening is the devaluation of humans. We are not giving gorillas the vote, or cows bank accounts. We are instead modeling human behaviour on animal behaviour, we treat embryos as if they were slugs, and the poor as if they were cattle. St. Thomas may have foreseen such problems, for the only way to bring animals and humans into equality is to eliminate that which is uniquely human.