Part of their argument against Muslims having a different God is that Muslims are Unitarian--meaning they believe there is only one Person in the Godhead, that is no Trinity, and deny the Divine Nature of Christ completely, regulating Him to a mere prophet. This is true about Islam, but does that mean they have a different God because they deny the Trinity and Deity of Christ? Protestants will rarely state Jews have a different God, since they tend to have less animosity towards them than Muslims (and understandably!). Here is roughly how the conversation went before I was kicked out of the room, it will be a paraphrase, since the text was lost when I was kicked:
Matt Slick: The Catholic Church is wrong in saying Muslims and Christians have the same God.
Me: Jews have the same God as Christians, and Jews say their God is the same as Allah.
Reformer*: Maimonides said Muhammad was a 'mad man!"
Me: Calling their prophet mad doesn't mean he was an idolater! He said Muslims were NOT idolaters even.
Since I was kicked out before I could provide evidence for the claim. Let's take "Reformer*" usage of Maimonides, despite him calling Muhammad 'al mashugah' he said Mulims were not idolaters (a term he had no problem applying to Christians).
...When your teacher called you a fool for denying that Muslims are idolaters he sinned grievously, and it is fitting that he ask your pardon, though he be your master. Then let him fast and weep and pray; perhaps he will find forgiveness. Was he intoxicated that he forgot the thirty-three passages in which the Law admonishes concerning "strangers"? For even if he had been in the right and you in error, it was his duty to be gentle; how much more, when the truth is with you and he was in error! And when he was discussing whether a Muslim is an idolater, he should have been cautious not to lose his temper with a proselyte of righteousness and put him to shame...--Maimonides Letter to an Inquirer (found in A Maimonides Reader, p. 477)
So this lays to rest any claim of Maimonides. For more information on the Jewish view Allah is the God of the Hebrew Bible read here (Jews praying in mosques/Allah is God), Jewish view on Islam
So the argument goes like this
P1: Jews have the same God as Christians.
P2: Jews believe Muslims have the same God as Jews.
Conclusion: Christians have the same God as Muslims.
Of course, some more refined Protestants will bizarrely insist Judaism is idolatrous because they deny the Trinity, and therefore worship a false god. But this is rare.
Matt Slick responded to my claim that Jews deny the Trinity like Muslims, he answered that Jews do not deny the trinity, they are just ignorant of it, whereas Muslims explicitly reject it. This is dishonest, if anything, Jews have lived in Christian lands for almost 2000 years, why should we think they are merely IGNORANT of it? Maimonides knew Christians were Trinitarian and insisted that makes them idolaters. This appeal to ignorance is awful.
Many of the Protestants insisted John's epistles state that if you do not have the Son, you do not have the Father. This is not a statement about believing in the true God, its about being IN God, and God in you. No one claims Muslims (or any group) are saved simply because they have the same basic concept of God (One Almighty, omnipresent God) and certainly rejection of Christ is a condition for damnation, a person cannot have the Father without also having the Son. They have to mangle St John's intent make this argument work!
Furthermore, the Apostles never accused the Jews of becoming idolaters for denying Christ, but simply "cut off from them people" and even "playing the harlot" by no embracing Christ.
During the Arian and modalist controversies the Catholic party never accused them of being idolaters or worshipping a false god because of their heretical understanding of the persons of God. Islam is a form of Arianism. The Church Fathers at the rise of Islam and after, as well as Muslim sources, despite Muhammad as being instructed by a Christian monk named Bahira (Buhaira in Arabic) who the fathers believed filled Muhammad's mind with Arian heresy. The church father did not accuse Allah of being a "moon-god," though they did accuse Muslims of inadvertently venerating the pagan goddesses image at Mecca.
Also, it should be mention Martin Luther, seemed to possibly accept the idea that Muslims worship "one true God."