Since "I think therefore I am" is meant to be a foundation for reasoning, it precedes any real definitions of "I," "thinking" and "being." So I think it's really more of a set of definitions than a conclusion.

We have a noun, "thought," which we define very broadly so as not to require any other definitions, and another noun, the self, which those thoughts are assumed to belong to. I think this is presumptive; working from first principles, why must a thought have a thinker? The self is a really meaty concept and Descartes just sneaks it in there unremarked-upon.

If you take that out, all you get is "thoughts exist." And even then, we're basically pointing at thoughts and saying "whatever these are doing is existing." Like, does a fictional character "exist" in the same way a real person does do? No, I think it's safe to say it's doing something different. But we point at whatever our thoughts are doing and define it as existence.

So I don't think we can learn much about the self or consciousness from Cartesian first-principles reasoning.