In other words, is an organism's ability to ignore some of the sensory input it receives from the outside world the first requirement in defining self from other?
Most perception theory does not think awareness works that way. Contrary to notions that come from more mystical traditions and got codified and popularized by psychoanalytic theories, it does not seem like we take in a broad spectrum of data and filter it trough some matrix to decide what will become conscious.
Instead, we keep a huge, parallel, unconscious model of the possible states of the world running, and we point-test its agreement with reality by addressing the senses to specific things. We keep a whole 3D model of our immediate surroundings, but we see only a tiny bit of that at a time, and our eyes dart about several times per second and check specific, unconsciously chosen pieces of it. We keep the model up to date, but it is not being actively perceived, and stuff that goes on can be totally missed, if the process that decides what it is most important to keep up-to-date focuses too narrowly.
From that theory, it is not true that we somehow take in a large part of everything around us. We take in a lot more than we are conscious of, but it is still preselected by that update process and not filtered out of a broad incoming stream that is passively received. Nor is it likely beings ever worked that way.
We are goal-directed beings at our core, and not as a consequence of some later separation from the world around us. Even bacteria are goal-directed beings that take in only relevant information. We are instantiations of a selfish genome made of selfish genes. It is more likely that we always functioned as individuals with agenda, separated from reality, and evolve in the other direction, from individual cells with agenda into cooperative colonies with a shared identity.
We did not separate from a broader context, we merged into a being. Consciousness concentrated from a dull awareness of our shared agenda into a centralized authority that contains the combined agenda and prioritizes its entries. As Dennet puts it, our bodies are made of cells in cells, prisoners to the central authority, but each in some tiny way also pursuing its individual agenda. Our cells evolved to survive independently for thousands of times longer than they evolved to function collectively, and things like cancer and autoimmune diseases are proof that they are not done learning to work together properly.
There is still a larger volume of unconscious data than conscious data, but the more likely reason for the separation is not choice, but lack of capacity. We have installed a central auditor of our process, and it attends to whatever part of the world it can manage. But we are also intensely social beings, so we have also structured this auditor in a way that makes it easiest to take its observations and share them with other similar animals. This severely limits its power.
We talk to ourselves, (we talk to our dogs) we constitute scenes with named objects, we keep our arms and faces uncovered so we can point and shout or emote visibly. All of this structures consciousness in the most sharable way, but limits its use of processing power. Interaction with others cannot reasonably take place completely in parallel. It has to be serialized. We think in story terms in case what results should be a story we need to tell someone else. Since the unconscious processes are free to run in a highly parallel manner, consciousness can only accommodate a small fraction of them, and it has to choose what arising phenomena are most important.
From that collection of perspectives what generates consciousness is limited ability and not active ignorance.