id love to know where you read that wonderful piece of information? it certainly wasnt in the koran. The same argument can be used against christianity, there is evidence and theories (more supported than anything i have found regarding islam) which points to many more books that have not survived to the bibles current incarnation, as well as other discrepencies with "truth" of the bible.
Judaism does not follow the model of the Trinity either, so is the Jewish god not the same as the christian god? Christianty still follows much of the old testament, and if you take the time to read the Koran, you will see that most of the old testament is repeated there. Like it or not, the theological figure of God and Allah are based on the same history, and both tell the same story. To say Islam doesnt share the same God is to say that Judaism doesnt share the same god, and if that is the case, then you must in turn deny the Old Testament, and the very basis of your own religion.