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