Christain God vs Muslim God
Are the two Gods the same God?
There are philosophical questions about reference lurking in that question that I'll pass over.
I guess that both Christians and Muslims identify their own gods with the earlier god of the Jews. That god is the capitalized "God", and is the (so they all say) the one single God that exists. (That's why I'm often uncomfortable with capitalizing "God", since doing so implicitly privileges the Judeo-Christian-Islamic family of religions.)
Christianity and Islam recognize much of the Hebrew scriptures and have absorbed Jewish mythology as their own. So historically, it's kind of like a three-pronged fork, with Christianity and Islam branching out of the Jewish mythological root and setting up independent existence, proclaiming that they have received additional revelations that Jews don't acknowldge.
While both Christians and Muslims accept much of the Hebrew scriptures, they proceed to reinterpret them dramatically to make them consistent with their divergent later views.
Christians read many Old Testament passages in ways that Jews consider out-of-context, interpreting them as if they were prophecies of Christ to come.
Muslims think that both the Jews and the Christians misunderstood and consequently screwed up God's primordial revelation. They see themselves as basically restoring the religion of Noah and even of Adam and Eve, wihout all the Jews' ethnocentric 'chosen people' distortions and the Christians' polytheistic proclaimation of Christ as a salvational god.
If so why do we appear to behave as if each group has a "monolpoly" over a particular version of the same God?
Christians and Muslims both believe that they are the recipients of God's final-for-all-time revelation. It's just that the Christians think that it was the death and resurrection of Jesus Christ, and the Muslims believe that it was God's revelation to Mohammed.