Muslims worship the same God as Christians and Jews do. They don't worship some different deity whose name is "Allah" -- "Allah" MEANS "God." Arabic-speaking Christians refer to "Allah" just like Muslims do. The Catholic Maltese (who speak a language directly related to Moghrabi Arabic) say "Allah" as well.
Didn't know that about the Catholic Maltese. Didn't know there were Catholic Maltese. (I do know where Malta is, though. I think.) The Maltese, however, Catholic or no, are not our subject today, but instead the identity of God Himself.
Remember a while back, when the President was explaining how he was ticked off, not at Muslims per se, but instead at the small minority thereof who did mean things like blow people up? Not surprisingly, in so doing he said something very much like the first sentence in the above quotation. As I recall, he was immediately challenged on this point by (I think) Richard Land, the head of the Southern Baptist Convention, who said (paraphrasing freely here) "No, they don't either, you big dummy!" So, which is right?
In this sense, I believe, God and/or Allah is not unlike a certain twice-baked yummy Italian cookie. Don't click that link yet! Let me explain first. Provoked by a claim about biscotti (i.e., that they aren't really biscotti until they've been cooked twice), in the linked post (after some dithering) I ended up saying, pretty much along Austinian and/or Wittgensteinian lines, that it doesn't matter which we say, as long as we don't get confused. It's not true, as some people claim, that the ontological facts of the matter swing completely free of our linguistic practices (that is, what we actually say), such that when speaking normally we might get (what we like to call) the appearances right (thus allowing for ordinary mistakes) but the underlying metaphysics wrong.
In our case, my immediate sympathies, ontologically speaking, were with the President (and Christopher S.) rather than Rev. Land (if that's indeed who it was). Here's why. A lot of people don't know the first thing about Islam (for some reason they don't teach this in school), and it is indeed worth pointing out that Islam is what we call an Abrahamic faith. Briefly and crudely, the Muslim story is that the God they worship ("Allah") is the God of Abraham the patriarch and Moses the lawgiver, as discussed in the Hebrew scriptures. Like Mohammed (peace be upon him), Jesus of Nazareth was a human being; a prophet (rasul), in contact (of some sort) with the divine, but apart from that, a human being like any other. Unlike pagans, Jews and Christians are "people of the book" (the Bible), and deserve at least that much respect. So when Muslims say (pardon my transliteration) La allahu ill' Allah ("there is no god but God"), they are agreeing with Jews and Christians that the God of Abraham is the only deity.
Naturally, about the nature of this entity, there remains some disagreement, which can get quite heated at times. So on this (Kripkean?) view what we should say is this: that the three faiths worship the same (i.e., numerically identical) entity, about Whom they believe different things (such that the three objects of devotion are qualitatively distinct). But now, as upon further reflection I have come to see, we run into the same issue as we did in the kitchen (with the cookies). Rev. Land's point was that the President was being silly to imply that the shared heritage, which of course he does not dispute, means that Muslims are somehow okay, theologically speaking. (Again I paraphrase.) The heart of the matter is this: Muslims reject the divinity of Christ. So what if Allah is numerically identical with the God of Abraham? In a way that makes it worse: Muslims aren't simply pagans (worshiping some entirely other, presumably non-existent, god), but heretics and blasphemers (worshiping God, supposedly, but getting his nature completely wrong: worse than Protestants (if you're Catholic) and worse than Catholics (if you're Protestant).)
In other words, the God Muslims worship is not triune, and did not send his Son to save us, which sounds like a different God indeed from the Christian God. Given this, it seems like the only reason to insist on marking the difference by referring to the two as numerically identical entities with different properties is the Kripkean one, that when the Prophet recited the Qur'an he referred to its ultimate source, naturally enough, with the Arabic name ("Allah") with which the God of Abraham had already been "baptized" (in the Kripkean sense; see Naming and Necessity for the whole story). But as far as I'm concerned the Kripkean story is entirely optional, best suited for hardcore metaphysical realists and their benighted ilk. For English speakers (and Arabic speakers), on the other hand, it seems perfectly natural to speak as Rev. Land does -- that is, to make the point about the difference between Muslims and Christians by speaking of the respective objects of devotion as (numerically) distinct entities. For Muslims, while Christians are indeed "people of the book," their "triune" God (whatever that means) doesn't exist; that would be blasphemy (or it would if it made any sense). La allahu, in other words, ill' Allah.
My point here, obviously, is metaphysical cum semantic rather than theological; on both (semantic) accounts the theological differences are the same, mutatis mutandis. On the other hand, however, particular differences may indeed only emerge on one way of speaking rather than the other: are Muslims firing their prayers into the void, or instead at a God against whom they blaspheme? What it is right to say, I claim (again putting the "facts of the matter" -- i.e., whether Islam or Christianity is the one true faith -- to one side), does not depend on an underlying ontology. The reason this is not idealism is that what it is right to say does depend, in part, on how things are (i.e., construed without the metaphysical appearance/reality dualism). If you want to hear more about cookies, or Austin, you can click that link now.