I was raised Catholic and believed in this concept for most of my life. Why do you think there is such a dichotomy in how the (supposed) word of god is interpreted by different sets of believers in both faiths? Taking out the human element, do you think its down to god in singularity, imparting a different way of 'being' to different tribes, or is it completely down to individual and tribal human perception in your opinion? Having read the bible growing up, and reading some of the Koran and examing the tenets of both religions, I simply cannot see how they can both be talking about the same deity. Just wondering what your opinion is on this.
These differences in faith not only exist between the three Abrahamic religions, but they exist within their own specific faiths. Eastern Orthodox, Protestants, Catholics all believe in different stuff. Even among the Protestants you have quite a lot of difference.
Moreover, if you have to make the argument that Islam does not worship the same God that Christians do, then, you should also make the same argument regarding Christians and Jews.
Finally it's about faith, you don't have to believe it, but they do believe it. Correct me if I am wrong, I bet you could find Jews who say the same thing about Christianity, that God isn't Yahweh, like some Christians say Allah isn't God.
If we're going to explain the differences, you have to approach it I think in at least two different ways. First, from the theist point of view. We will assume God is in fact real, and that these holy books do in fact represent the word of God. In which case, what it would mean is, God has many faces, there is no one path to God, and God shapes its message for the individuals receiving it.
If we approach it from the point of view of someone who isn't religious, then we can introduce things like culture, and use cultural archaeology to explain why things are the way they are, like Arabs codifying veils into the religion, because it was part of desert survival. Then we can look at other human traits, like greed, power, corruption, and understand that these holy books are not the word of God, rather they are the word of men looking to enforce control over other men, and a great way to do that, is to piggy back on a pre-existing set of beliefs that are already widely popular.