No. The god they worship is, in reality, a demon. In terms of their own theology, however, he is a single person, transcendent. The God we worship, on the other hand, is the maker of heaven and earth. He is one being and transcendent, who exists in three persons, which are also immanent. Neither the one-ness nor the three-ness of God are tangential attributes. They are instead essential attributes; they define who He is. Move away from that definition and you move away from the true and living God.
That same God has spoken in His Word, the Bible. He has not spoken again in the Koran. The God we worship did not send his most important prophet hundreds of years after the ascension of Jesus. The differences are too many to list. The overlap is here- in both instances we have a creator, a unity, a judge, one who transcends, who gives law, who is called “God.” (Allah is not a distinct name for God, but is simply the Arabic word for “God.) To suggest that they are the same would be like arguing that since my sister and I both live in central Florida, both have red hair, both work for the same organization, both have the same parents and both think my wife is wonderful, that we are in fact the same person.
The better question is this- given the obvious and yawning gap between who God actually is and what Islam says their god is, why would anyone try to pretend that they are the same? It could be for the sake of earthly peace. At least some people in both faith groups are weary of all the killing. If our killing of each other, which includes ongoing wars in the middle east, ongoing attacks on our own soil, all the way back to the Crusades, is grounded in different views of who God is, and if we can agree we’re both right, then we can all lay down our arms. Or it could be a ruse in the ongoing battle. If Islam can get the Christians to lay down their arms, they can win more victories. If Christians can get Muslims to relax, maybe we’ll be able to tell them about Jesus. We build bridges for evangelism. They build bridges to attack. My suspicion is that we, on the Christians side, affirm such nonsense for a very simple reason- we don’t believe. This kind of nonsense is not the result of succumbing to Islam, but on both sides, succumbing to the unbelieving spirit of our post-modern age.
The Bible teaches clearly that those who do not trust in the finished work of Christ alone will spend eternity in torment. To deny Jesus, His incarnation, His perfect life, His atoning suffering and death, His resurrection and ascension to the right hand of the Father is to be damned. Islam does just that. Islam not only does not look to the work of Christ for atonement for sins, they have no doctrine of grace at all. One cannot worship the true and living God while denying His Son. If the Bible’s message here is true, it is false that we worship the same God. And we are in grave danger to suggest otherwise. Islam is a false religion, created by a false prophet who gave the world a false book. The Christian faith is true, because Jesus is the Word, and His Word is true.