What is the basis for saying Christians and Muslims worship the same God?

The understanding of what God is is unique to each individual. God is to be experienced, and cannot be described, or truly known. The answer to your question of whether or not we understand our God to be the same can be found from our own personal experience of God, compared with that of a Jew, and a Moslem. The question addresses the worship of God - and obviously the various monotheistic Religions have different methods of worship, which we abide by. Even though we worship differently, it does not imply that our God is in any way different though some Religions do teach this.

In the ACC the Doctrine of the Apostolic Churches was formulated before the 6th. cent. from the decisions of the Bishops during the first seven Ecumenical Councils, and this before the Muslim Religion was established, so there is no reference in the Church to any Moslem understanding of God. The ACC does teach that the Jewish and Christian God is the same and God is written as the tetragrammaton YHVH (or YHWH) in the Hebrew writings. In Judaism when YHVH is read, the word spoken is Adonai. In Christianity, the word spoken is Yahwey, or Jehovah. None of these words occur in the NT. So, many Christians are unclear as to whom The Father of Jesus is in relationship to God YHVH - what is, in fact, the theological definition of The Father? And, today, how does this relate to the Moslem idea of God? This, I think is where the confusion lies.

During the first three centuries, the Bishops were unsure of how The Father and the God of Judaism were related. For perhaps a hundred years after the crucifixion, most Christians taught that The Father was not the same as the God of Judaism. Some people believe this today. But the catholic Church (the universal Church of East and West,) eventually stated that the Christian God was the same as the God of Judaism. They were forced into this in order to include Christians into Israel the children of God to whom the covenants were given. At that time (when the statement was made,) the writings of Paul were influential in the Church, in which Paul explicitly says that Christians are grafted into the vine of Judaism, so that the promise of God to Israel also applies to Christians. And warns that if Israel does not gain the promise, then neither do the Christians, and all will be lost. If, then, Paul is correct, we Christians share the same God as Judaism. And since Moslems also claim to be people of the book (Torah) and share the same God, we can be reasonably sure that we all share the same God.

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?

I hold that Christianity is defined by what the Bible teaches and therefore I take the position that Muslims (and Jews) don't worship the same God as Christians. They reject the Trinity and believe that Jesus wasn't the Son of God who is one with the Father and Holy Spirit. They therefore reject God's revelation of Himself which is found in the Bible. So my position as a Christian is that Muhammad wasn't a true prophet and the religion he authored isn't divinely inspired. This was also Luther's position who said the following (for “Turks” read “Muslims”):

“Turks and Jews boast much of God and profess to have a better faith than we Christians. They say they cannot be wrong; for they believe in the one God, who created heaven and earth and everything. A faith such as this can certainly not be wrong, they think. Christ, however, here concludes: He who hates Me hates My Father. Now since Turks and Jews hate Christ and persecute His Word, they certainly also hate the God who has created heaven and earth, do not believe in Him, and do not honor Him. For Christ is this same one God.” (W 52, 731 —E 3, 498 —SL 13a, 1285)

The first question is:

Muslims claim that they worship the same god as Christians do. . . . Are there any Christian doctrines that accept this claim?

Yes.

The Christian doctrine of Emanuel Swedenborg (1688–1772) accepts the claim that Muslims worship the same God as Christians (and Jews and everyone else who worships God).

Though Swedenborg had some faulty ideas about Islam based on common 18th century European notions of Islam (such as the idea that Muslims could accept Jesus as the Son of God), his overall position was that Islam was founded under God's providence in order to root out the idolatry common in the Middle East at the time of Mohammed, and to establish in its place the worship of one God in a way that is adapted to the cultures of the Middle East.

About Muslim non-acceptance of Jesus as God, he wrote:

The reason they did not recognize the Lord [Jesus Christ] as the God of heaven and earth was that the people of the Middle East believed in God as the Creator of the universe and could not grasp the idea that he had come into the world and taken on a human nature. (Divine Providence #255)

His general teaching about people of other religions besides Christianity is this:

People born outside the church are just as human as people born within it. They come from the same heavenly source. They are equally living and immortal souls. They have religions as well, religions that enable them to believe that God exists and that they should lead good lives; and all of them who do believe in God and lead good lives become spiritual on their own level and are saved. (Divine Providence #330)

In Swedenborg's view—and in the view of the members of the various Christian denominations that have been founded based on his theology—Muslims worship the same God as Christians because there is no other God to worship.

The second question is:

If so, what is the biblical basis for this claim?

Since Islam didn't come into existence until several centuries after the last books of the Bible were written, obviously there is no mention of Islam in the Bible, and therefore there are no statements in the Bible about whether Muslims do or don't worship the same God.

Once again from a Swedenborgian perspective, though, the answer to this question is very simple:

The Bible states clearly, a number of times, that there is only one God. See for example, Isaiah 45:21:

Declare and present your case; let them take counsel together! Who told this long ago? Who declared it of old? Was it not I, the Lord? There is no other god besides me, a righteous God and a Savior; there is no one besides me.

Since according to the Bible there is only one God, all people who worship God (as compared to worshiping idols, deceased human beings, and so on) are worshiping the same God, because there is no other God to worship.

The fact that Muslims attribute some characteristics to God that Christians don't, or that Christians attribute some characteristics to God that Muslims don't, is not as relevant as traditional and conservative believers in either faith think it is. After all, even within each of those religions there is a wide variety of opinion and perspective on the exact nature of God.

The old Indian tale of the blind men and an elephant nicely illustrates the idea that even if our ideas about God may differ due to our different cultural and religious perspectives, it is still the same God.

Here is Swedenborg's succinct way of saying this with regard to belief in the Jesus, later on in Divine Providence #330:

The Lord [Jesus Christ] is known to everyone who believes in God because the Lord is the God of heaven and earth, as he tells us in Matthew 28:18 and elsewhere.

Matthew 28:18 reads:

And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. (KJV)

As Swedenborg points out elsewhere, "all power" is divine power, or the power of God. So if all power in heaven and earth is given to Jesus, then Jesus is the God of heaven and earth. (It makes no difference if you translate it "all authority" as many modern translations do. All authority is still divine authority, which is the authority of God.)

Short version: From the perspective of Swedenborgian doctrine, the biblical basis of the claim that Muslims worship the same God as Christians is that according to the Bible there is only one God, so everyone who worships God is worshiping the same God.

Muslims may not appreciate (or perhaps they would) this distinction. But Hilare Belloc claims in The Great Heresies, that Islam is a heresy of Christianity.

Therefore, if Mohammed had had a vague inkling of Christianity and claim the inheritance of Ishmael then yeah, of course they worship the same God.

However, we couldn't say they worship Him fully because they do not believe in the Trinity. They do not believe that Jesus could be God.

In response to Caleb's answer wherein he says that no Christian sect, except liberal traditions embrace Muslims as believing in the same God.

From the documents of Vatican II (not just something a talking head says):

The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth, who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.

Nostra Aetae - From the Documents of Vatican II

With reference to Muslims, it states these few things which are common to our faith and conveniently takes a distant reference from Abraham which is a common point on worshiping same God. It makes no reference to Mohammad though he is the epitome of this religion. Again, no recognition of Muhammad, the Qur'an, and the correctness of what Muhammad taught. So, as becomes clear in other Church documents, whatever truth there may be in what Muslims believe must come either from the universal truths that God has always made accessible to the minds and hearts of all people, or else by transmission from the divine revelation granted to the Jews and Christians.

So, yeah, even though this questions wasn't directed at Catholics in particular, it is true that most Christians (and the entire Communion of Saints) believe that Muslims attempt to worship same God, but they do so in severe error.

and, just so no one thinks this is out of context, the document says of Islam and other non-Christian religions

The Catholic Church rejects nothing that is true and holy in these religions. She regards with sincere reverence those ways of conduct and of life, those precepts and teachings which, though differing in many aspects from the ones she holds and sets forth, nonetheless often reflect a ray of that Truth which enlightens all men. Indeed, she proclaims, and ever must proclaim Christ "the way, the truth, and the life" (John 14:6), in whom men may find the fullness of religious life, in whom God has reconciled all things to Himself.

My pastor explained it pretty succinctly to a Religious Ed. class who was doing some Q&A with him. He said it's like a theatre where everyone's looking at the same play but quite a few people (non-Catholics and non-Christians) have for whatever reason decided to watch the play from awful seats instead of getting the front row seats that they could have had, had they only desired them. That's why I said above, they do not worship Him fully, but they do worship Him.