I think you hit on two things of which we can pretend we have a cursory understanding. In your eariler section, I think you're calling up Heidegger's da-sein, the being that is concerned with its own being. As much as I'm enjoying "languification," we may not need that yet. I think da-sein is code for the human condition you describe, whereby we ask "what am I?" And of course, "what are you?" In that sense, I think we're not so much in the realm of phenomenology, but in basic existentialism.
We've already played something like that identity game in ethics, when Nietzsche suggests that any promise ("I will shovel the snow tomorrow") hides the unenforceable promise that my identity will not change ("I will be the same person tomorrow"). So really, if we accept this premise, then the teleporter and Spontaneously Occuring Lincoln arguments are an attempt to extend that promise paradigm (identity in some mental space) to question physical identity.