Orthodox Christians, whether Catholic, Protestant, or Eastern Orthodox, have a different conception of God's nature than do Jews or Muslims, or heterodox groups like Unitarians, Mormons, and Jehovah's Witnesses. The clearest definition of the orthodox Christian concept of God may be found in the Athanasian Creed. "And the catholic faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the persons, nor dividing the substance. For there is one Person of the Father, another of the Son and another of the Holy Spirit. But the Godhead of the Father, of the Son, and of the Holy Spirit is all one, the glory equal, the majesty co-eternal. Such as the Father is, such is the Son and such is the Holy Spirit."
This position is rejected by all of those groups beyond the pale of orthodoxy, but is based on Biblical teachings. The Jews do not accept the New Testament. The Muslims regard the Koran as perfecting and superceding the Bible. The Mormons regard The Book of Mormon, the Pearl of Great Price, and Doctrines and Covenants as revelations equal or superior to the Bible. The Unitarians deny that the Scriptures are divinely inspired or without error in matters of faith and doctrine. The Jehovah's Witnesses use a translation of the Bible that confounds both traditional and modern Scriptural scholarship. Hence, all these groups lack the grounds in their sacred writings to support the orthodox Christian definition of the Trinity.
The same could be said for the orthodox Christian definition of the nature of Jesus Christ, as defined by the Council of Chalcedon. Jesus Christ is both truly God and truly man, being "recognized in two natures, without confusion, without change, without division, without separation; the distinction of natures being in no way annulled by the union, but rather the characteristics of each nature being preserved and coming together to form one person and subsistence." All those beyond the pale of orthodox Christianity, whether Muslim, Jewish, Mormon, Unitatian, Jehovah's Witnesses, or otherwise, understandably reject this doctrine, though Scripture supports it.
Christians of orthodox belief disagree with the aforementioned other groups as to the nature and character of God, even if they agree with them on His power and attributes. Therefore, it cannot be said that orthodox Christians worship the same God as the others.
President Bush is perhaps playing to a popular "unity in diversity" theme, sort of a revival of the old saw, "the fatherhood of God and the brotherhood of man." He is practicing a type of American civil religion that was expressed by Dwight Eisenhower 50 years ago in words to the effect that our political system was founded on a belief in God, but that he didn't care whose God it was. Bush's statement and that of Eisenhower may make political sense, but they do not make religious sense.
Or they do as a expression of liberal Protestantism mixed with the syncretic deism of the lodges. Faced with the fierce zeal of real Islam this wishful thinking turns into absurd.