The Gospel Coalition has kindly asked me to write on this subject, and the post is now up, here.
One of the commentators opined that the "Clark Kent/Superman" argument is not intended by those who put it forward as a positive argument that Muslims and Christians do worship the same God but only as an answer, which he considers a good one, to the argument that Christians and Muslims don't worship the same God because of the major differences in doctrine and concept of God between the two religions. I wanted to highlight my response to that, here, and then (if this is brought up again), I can always link to it again if this question comes up again. To his claim that he isn't hearing people using this as a positive argument, I reply,
Well, then you haven't "heard" the same things I have "heard." It is definitely being used as a positive argument in various spontaneous debates all over the place (e.g., on Facebook), and it is being written up in ways that strongly sound that way even when someone might try to claim "deniability" that that was the intent. See, for example, this article by Beckwith, where the Clark Kent/Superman analogy is *quite reasonably* interpreted as a positive argument, though he may say that it was not his intention.
Moreover, you are simply wrong to say that this would be a good response to arguments about divine properties and differences of concept. The arguments about the divine properties, especially essential properties, create a strong prima facie case that the two deities are not the same. The Clark Kent/Superman analogy does _nothing_ to answer this prima facie case. It merely points to a possibility. Who cares about bare possibilities? How do bare possibilities refute the perfectly legitimate prima facie case from major differences of concept?
Notice that this would be the case with even the scenario with Clark Kent himself. If someone proposes to Lois Lane (without positive argument) that Clark Kent and Superman are the same person, it's completely reasonable and relevant for her to reply that Clark Kent is dweebie, shy, wears glasses, and shows no signs of super-powers. That's a case that has to be answered if someone wants her to _believe_ that they are the same being. It is _no_ answer to that case to say, "Hey, it's possible that Superman is Clark Kent's secret, superhero alter ego." That is not a "good refutation" of Lois Lane's argument. After all, there's a reason why Superman's identity with Clark Kent _is_ generally thought of as a secret that most people don't know--Superman goes to some trouble to keep the identity quiet and to make them appear different.