One of the claims which I saw come up in a number of discussions of this among Catholics, however, is: Muslims don't even worship the same God as we do.
Father Longenecker wrote a fairly good piece responding to that claim. His argues that the fact that someone's understanding of God is defective or incomplete does not mean that person is not worshiping "the same God". There is, after all, only one God:
The real heart of your question is that you can’t get your head around the idea that Muslims worship the same loving Father in heaven that we do.One can see Fr. Longenecker's affection for C. S. Lewis coming through here, as this line of thinking will be very familiar to readers of Lewis's The Last Battle. It's not a bad way of looking at things, but I think there are a couple other elements of this which are worth thinking about.
It’s a problem you share with many Muslims who are horrified not only at the Christian idea of the Holy Trinity (they think it is polytheism) but they are also very resistant to the idea of God as Father. Allah is completely transcendent and any idea that we impose human characteristics on to him they find abhorrent.
So how can we say that they worship the same God?
Simply because there is only one God.
If someone worships God as he conceives of him he is worshipping the one, true God.
In the sense that Fr. Longenecker is speaking, someone who sincerely worships Zeus is worshipping God. In the sense that there is no Good but God, I think that's clearly true. But it is still possible to worship false Gods. Whether one holds that Zeus simply didn't exist, or that he was some sort of non-human being who was not God, it was certainly possible to worship him and in worshipping him, one was worshipping something other than God.
Is Allah a false god?
It seems to me that we have to say "No." Muslims worship the one God of Abraham. They take the Old Testament of our Bible to describe God's relationship with man. They believe (wrongly) that Jesus was a prophet rather than God. But then, Jews do not believe that Jesus was God. If we hold that Muslims do not worship "the same God" because they do not believe in the Trinity, then we have to hold that Jews do not worship the same God either -- and that's rather problematic given that we believe God Himself came to earth as an observant Jew.
Another concern I've heard expressed is that Islam stems from the revelation which Mohammad allegedly received from God via the Angel Gabriel. Obviously, as Christians do don't believe that a "revelation" which denies the divinity of Christ and our salvation through His sacrifice on the Cross is a true revelation. With the end of Apostolic times came the end of general revelation. There will, if Catholicism is true, be no more prophets. So, if this revelation which we must consider false is the source of Islam, and God would never have sent a false revelation, must this not mean that Muslims do not worship the same God?
I don't think that necessarily follows. Clearly, as a Catholic, I do not believe that the Koran is the word of God. That means that I would have to conclude that Mohammad was either either deceived or lying when he claimed to receive a revelation through the Angel Gabriel. If deceived, this could be either the result of some kind of natural delusion or a result of some real supernatural creature (but not one of God's faithful angels) giving him a false message.
However, it's entirely possible to tell an untruth about a real person. By telling an untruth about a person, we don't somehow create an alternate, false person. We just tell a distortion of the truth. Even if we take the position that Mohammad was spoken to by a fallen angel, this would not mean that Islam worships some other God. There is, after all, only one God who revealed Himself to Abraham, and this is the God that Islam names as Allah. Lies always exist in reference to truth, they don't exist on their own. If the father of lies or one of his servants was the source of Mohammad's revelations, what could be more mischievous than to base his lies in the obvious truth of God's revelation of Himself to Israel and the Incarnation?