I'm on Paul's side here, but let me offer some detail. I think we need to be more careful here. On what basis do we decide if two entities are the same thing? In other words, we need to agree on a theory of names before we can answer this question.
A common view is what is called a descriptive theory of names. On [url=
http://plato.stanford.edu/ archives/spr2010/entries/reference/]this view, “a proper name . . . refers via the descriptive content associated (by the speaker) with that name. This descriptive content is thought to uniquely determine the name's referent." So "the Christian God" would be something like "The Triune Being that sent Jesus into the world to die for sin," whereas Allah would be something like "the God that commissioned Mohammad to take His message into the world."
Now, I reject that view for reasons I won't go into here. My own view is that names are words that represent ideas or concepts, and that ideas or concepts are
necessarily associated with certain realities. So suppose I have the concept of a tree. I attribute certain things to it. It is made of wood, has leaves, is a plant, etc. Now suppose an ancient Israelite thinks of the same thing I am. He uses the word
ets to describe it. Is
ets the same thing as "tree"? Yes, because the words reflect identical concepts. But now suppose that I am thinking of this tree, and yet I hold to some wrong ideas about it . . . suppose I think, for instance, that the tree has smooth bark whereas my Israelite friend is thinking of one with rough bark. Are
ets and "tree" still identical? Yes. They are. I just may be wrong in my assertion that all trees have smooth bark, or that this particular tree does. In other words, I can have some wrong ideas about a concept (that is, the concept may be imperfectly formed) and still be thinking of the same concept that you are. You may have a
better understanding of the concept than I do, but disagreeing on a concepts attributes doesn't mean that we are not thinking of the same thing.
So, when
do we say we are no longer thinking of the same thing? Take the word "Herd." I say that, and I am thinking of a flock of animals (say, of elephants). A German says the same word and he is thinking of what we would call a stove! So things words mean different things when the concepts they are attached to are different concepts. But what makes concepts identical? It can't just be a mistake about their attributes, because we've already seen that some mistakes don't qualify as changing the concept. They just mean that we have an unclear view of the concept itself. That is, we are talking about the same concept--one (or both) of us just are mistaken when it comes to some aspect of it.
I would suggest that the answer lies in comparing the things' essential properties. If they are the same, then the two concepts are identical. If they are similar enough that they exclude all other possibilities, then they are at least attempting to be the same thing. So suppose you and I are both thinking about a fruit and we want to know if the fruit we are thinking of is the same thing. I say that mine is essentially round. So is yours! I say mine is essentially sweet. So is yours! I say the peel is essentially smooth. Oh, no . . . yours is essentially fuzzy. It turns out that apples and peaches aren't identical after all. But suppose that your fruit is essentially smooth. But now suppose that I insist mine is essentially red and you that it is essentially green. Well, it turns out that our apples are identical (there are no other smooth, sweet, round fruits (etc.--add enough properties to limit this to a discussion of apples and not, say, grapes); we are both just mistaken. I mistakenly thought that all apples are red, and you mistakenly thought that all apples are green.
So this shows that there is an approximation of essential properties when it comes to trying to decide of two concepts are identical.
How does all this apply to Allah vs. God?
They clearly have many essential attributes in common. Both are divine. Both are the Creator of the universe. Both are to be worshiped, obeyed, etc. Both are eternal, exist
a se, are sovereign, etc. But there are things that they do not have in common. One is unitarian, one trinitarian. One was incarnated, the other not, etc. So are they the same or not?
I would suggest that yes, they are, insofar as we are both talking about the eternal, divine Creator of the Universe. The problem, then, is not that we are talking about different gods. It is that one group (or both) has made some false statements about God's nature. And this, I would suggest, is the biblical answer. In Job 42:7, God says, "My anger burns against you and against your two friends, for you have not spoken of me what is right, as my servant Job has." If wrongly attributing things to God means that we are no longer talking about the same God, then God could not say that they had "not spoken of Me what is right." They would have been speaking of someone else entirely (the god of their own creation!).
So, tl;dr - yes, we are talking about the same God. There is only one God to talk about. The bigger question is this: are we saying what is right about Him. When Muslims and Christians make contradictory claims, which (if either) is correct? Which, that is, is speaking what is right about Him?
edit:
And let me throw a monkey wrench into everything I said above. Suppose that Allah is actually the name taken on by a demon that deceived Muhammad. Then, at least, Muhammad would have been talking about the demon while thinking he was talking about God. For the demon would be shown to be impersonating God. So here, the Muslim is actually talking about a demon though he intends to be talking about God. So intention is not sufficient to establish identity. The more important question is the reality itself we are talking about. Thus, in addition to the philosophical analysis provided above, it seems a historical analysis can help answer this question as well.
In other words . . . it's complicated.
