"Many Christians object to Muslims saying "god is great" but are quite happy to say hallelujah which effectively means the same thing (and even refers to the exact same God)."
Wrong on ALL counts. Allahu akbar means "God is greater" not God is great. Meaning to Islam that Allah is greater than any other god.
"Hallelujah" is from the Hebrew word Hallel and is used to express praise or joy, the Greek form is Alleluia.
Allah is NOT the same God of Christians, The Quran describes Allah as the best deceiver there is, a liar who is not above using the same evil and wicked schemes of his opponents. For example, the Quran calls Allah a makr, in fact the best makr there is: But they (the Jews) were deceptive, and Allah was deceptive, for Allah is the best of deceivers (Wamakaroo wamakara Allahu waAllahu khayru al-makireena)! S. 3:54; cf. 8:30.
In the Catholic and Christian faiths God cannot be deceived nor does he deceive. "We believe 'because of the authority of God himself who reveals them, who can neither deceive nor be deceived'" (Catechism of the Catholic Church #156) The Muslim god Allah is NOT the same as the Christian God.