"A subject" is a universal way to say "I" because we can all agree on it. If I were to say "I observe myself", then you could argue with me because there is no evidence that "I" am anything more than a part of "You". The people in our dreams, for example, are extensions of ourselves, no? Perhaps I am an extension of you right now. Maybe this is a dream or somesuch thing.

But to put it more simply, the only "subject" that one can vouch for is themselves.