Most organisms, I think, are self-aware in this limited, situational sense (if they are aware at all). However, there is another very important difference between the awareness of many organisms and our own. We are aware of ourselves as unique, as an individual different from any other person or organism on earth. This is such an obvious and deeply-rooted belief that most views of self-awareness implicitly understand it as part of what it means to be self-aware.
But it is possible in principle to have a much simpler sense of self . For example, most organisms, even very simple invertebrates, behave in a manner that suggests they distinguish themselves from the environment. To the extent that they have any awareness, therefore, they should have the experience of themselves as something apart from the rest of the world. This does not imply any sense of being unique, however, for that requires multiple distinctions drawn between themselves and other organisms. Likewise, certain more evolved invertebrates experience themselves as members of a group. This is a higher sense of self, somewhat closer to our own, but it still lacks the experience of individuality, since the organism behaves more or less exactly like other members of its group.
I believe if we understand self-consciousness in these broader terms, we can plausibly extend the notion to most other forms of life, at least to those which we believe have any awareness of any kind. I discuss this in detail in my book The Dimensions of Experience (DE), from which this post has been drawn, with some modifications. I also argue at length that the consciousness of other organisms is intersubjective, though they lack language. To his credit, Wilber suggested this idea some time ago, though he never provided much of what I would consider real evidence for it. In DE I discuss at length how we are to understand intersubjectivity as manifested in other forms of life. [10:39 AM 12:12 PM 4:53 PM 3:16 PM 12:42 PM]