In response to the question posed by the original poster, "Is the God of Islam the same God as is worshipped by Christians?" it all depends on how one defines one's terms. Muslims confess that Allah is the transcendent creator of the universe, so I suppose in one sense
one might say that both Islam and Christianity believe in the same God. Muslims confess that Allah is the God worshipped by Abrahams, so I suppose in one sense
one might say that both Islam and Christianity believe in the same God. But ... the respective understandings of who God is is so radically different that one might also say that the two religions worship two very different deities.
I am reminded of this telling passage from G. K. Chesterton's Orthodoxy
Unitarians (a sect never to be mentioned without a special respect for their distinguished intellectual dignity and high intellectual honour) are often reformers by the accident that throws so many small sects into such an attitude. But there is nothing in the least liberal or akin to reform in the substitution of pure monotheism for the Trinity. The complex God of the Athanasian Creed may be an enigma for the intellect; but He is far less likely to gather the mystery and cruelty of a Sultan than the lonely god of Omar or Mahomet. The god who is a mere awful unity is not only a king but an Eastern king. The heart of humanity, especially of European humanity, is certainly much more satisfied by the strange hints and symbols that gather round the Trinitarian idea, the image of a council at which mercy pleads as well as justice, the conception of a sort of liberty and variety existing even in the inmost chamber of the world. For Western religion has always felt keenly the idea "it is not well for man to be alone." The social instinct asserted itself everywhere as when the Eastern idea of hermits was practically expelled by the Western idea of monks. So even asceticism became brotherly; and the Trappists were sociable even when they were silent. If this love of a living complexity be our test, it is certainly healthier to have the Trinitarian religion than the Unitarian. For to us Trinitarians (if I may say it with reverence) -- to us God Himself is a society. It is indeed a fathomless mystery of theology, and even if I were theologian enough to deal with it directly, it would not be relevant to do so here. Suffice it to say here that this triple enigma is as comforting as wine and open as an English fireside; that this thing that bewilders the intellect utterly quiets the heart: but out of the desert, from the dry places and, the dreadful suns, come the cruel children of the lonely God; the real Unitarians who with scimitar in hand have laid waste the world. For it is not well for God to be alone.
Apart from God's self-revelation in Jesus Christ and Israel, monotheism can be a very dangerous and destructive religion.