The answer depends on what is meant by "the same God", specifically whether it refers to What God is or to Who God is.

Let us assume that the world is populated by humans and centaurs, and that we have the works of Aristotle but do not know whether he was human or centaur. In this case, we know who Aristotle was, i.e. what he thought, but do not know what he was, i.e. whether he was a man or a centaur.

Regarding What God is, Christians believe in "one true and living God, Creator and Lord of heaven and earth, almighty, eternal, immense, incomprehensible, infinite in intelligence, in will, and in all perfection, Who, as being one, sole, absolutely simple and immutable spiritual substance, is to be declared as really and essentially distinct from the world, of supreme beatitude in and from himself, and ineffably exalted above all things which exist, or are conceivable, except Himself." [1]

Although this definition is by a Catholic Ecumenical Council, I am sure almost all Christians would agree with it (except for Palamist Eastern Orthodoxs holding a real distinction between Essence and Energies, who might object to "absolutely simple"), and fairly sure that most Muslims would too. In that sense, then, Muslims and Christians worship the same God, Who is the Absolute Fullness of Being and created and sustains everything outside Himself.

To note, while Maimonidean Jews would probably agree with the above definition, those holding Kabbalistic doctrine might not, specifically regarding divine simplicity (vs the sefirot) and immutability (vs tzimtzum).

Regarding now Who God is, it is clear that Christians and Muslims, or Christians and Talmudic Jews [2], do not worship the same God.

Addendum: After writing this, I found an answer to the same question which IMO is along the same lines by Dr. J Dudley Woodberry, Dean Emeritus and Professor of Islamic Studies at Fuller Theological Seminary [3].

[1] Ecumenical Council Vatican I, Constitution "Dei Filius", ch. 1 "Of God, the Creator of all Things". http://www.ccel.org/ccel/schaff/creeds2.v.ii.i.html

[2] I leave aside Karaite Jews, as they might be in the situation of the Jews before Jesus, and no Christian would say that Jews before Jesus worshipped a different God regarding Who God is, even though they had an incomplete knowledge of God as compared to Christians.

[3] http://cvi2.org/paul-timothy/pages/gm/woodberry_same_god_4_2007.pdf

The answer depends on what is meant by "the same God", specifically whether it refers to What God is or to Who God is.

Let us assume that the world is populated by humans and centaurs, and that we have the works of Aristotle but do not know whether he was human or centaur. In this case, we know who Aristotle was, i.e. what he thought, but do not know what he was, i.e. whether he was a man or a centaur.

Regarding What God is, Christians believe in "one true and living God, Creator and Lord of heaven and earth, almighty, eternal, immense, incomprehensible, infinite in intelligence, in will, and in all perfection, Who, as being one, sole, absolutely simple and immutable spiritual substance, is to be declared as really and essentially distinct from the world, of supreme beatitude in and from himself, and ineffably exalted above all things which exist, or are conceivable, except Himself." [1]

Although this definition is by a Catholic Ecumenical Council, I am sure almost all Christians would agree with it (except for Palamist Eastern Orthodoxs holding a real distinction between Essence and Energies, who might object to "absolutely simple"), and fairly sure that most Muslims would too. In that sense, then, Muslims and Christians worship the same God, Who is the Absolute Fullness of Being and created and sustains everything outside Himself.

To note, while Maimonidean Jews would probably agree with the above definition, those holding Kabbalistic doctrine might not, specifically regarding divine simplicity (vs the sefirot) and immutability (vs tzimtzum).

Regarding now Who God is, it is clear that Christians and Muslims, or Christians and Talmudic Jews [2], do not worship the same God.

Addendum: After writing this, I found an answer to the same question which IMO is along the same lines by Dr. J Dudley Woodberry, Dean Emeritus and Professor of Islamic Studies at Fuller Theological Seminary [3].

[1] Ecumenical Council Vatican I, Constitution "Dei Filius", ch. 1 "Of God, the Creator of all Things".

[2] I leave aside Karaite Jews, as they might be in the situation of the Jews before Jesus, and no Christian would say that Jews before Jesus worshipped a different God regarding Who God is, even though they had an incomplete knowledge of God as compared to Christians.

[3] DO CHRISTIANS AND MUSLIMS WORSHIP THE SAME GOD?