One could argue that self-awareness begins when we start learning language and attaching names to everything, including attaching names to ourselves. We are taught language by pointing at people or objects and naming them. I think the quote could easily be "I think using words/language, therefore I am."

I also get a weird feeling sometimes when I say my first name aloud, or say I am ______. Language affects how we see and understand ourselves. Our thoughts are in our language and everything and everybody is identified by a name, so if we take away language, are we also taking away meaning? How do you ask yourself "who am I" without language? I don't know if I'm making sense or not, but anyway.