do muslims profess to worship the same god?

 Catholic teaching: the muslims do worship the same god as christians.

Catechism of the Catholic Church 841:

“The plan of salvation also includes those who acknowledge the Creator, in the first place amongst whom are the Muslims; these profess [note: not actually, but only by profession] to hold the faith of Abraham, and together with us they adore the one, merciful God, mankind's judge on the last day."

Vatican II’s Declaration on the Relation of the Church to Non-Christian Religions, Nostra Aetate, 3:

“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.”

Argument: “ As an analogy, imagine a child who was adopted but didn't yet know it. He or she might say, “I am really thankful that my mother gave me birth.” Now, this person thinks that his or her birth mother is the woman who in reality is only the adoptive mother. But nevertheless, the attitude of thankfulness for having been given birth in a sense “transfers” over to the real birth mother [referent]” (Dave Armstrong’s analogy: https://www.detroitcatholic.com/voices/do-catholics-and-muslims-worship-the-same-god)

so too with the question of whether the muslims do worship the same god as christians. The Christians are right God is a Trinity and the Muslims deny that God is a Trinity. But they can still have the same referent of the God of Abraham or the monotheistic God.

Moreover: Or consider the fact that when we say that Muslims and Christians worship the same God—we are referring to the reference of YHWH Father as shown in the Old Testament, but NOT Jesus nor the Holy Spirit. This is true, even if Muslims deny that YHWH God is “Father.” As shown from the analogy above because both parties refers to the same referent

Similar to how Jews deny the Trinity but worship the same God according to early Church Fathers:

1. Augustine: "The Jews, it is true, worship the one omnipotent God, but they expect from him only temporal and visible goods" (For full context and quotation, see: Of True Religion, introduction by Louis O. Mink, translated by J. H. S. Burleigh [Henry Regner Company, Chicago, Illinois, third printing 1964], pp. 8-11)
2. Gregory of Palamas: "Then those who were set to do the burying take the box up on their arms and walk further down. All the rest of them, with the Tasimanes, returned home. As I was sitting there I asked whether anyone could speak both languages that I needed. There was somebody, whom I asked to say TO THE TURKS [Muslims] on my behalf that what they had performed outside there I thought it was good, “for you addressed yourselves to God – to whom else? – for the deceased one. I wanted, however, to know what was that you exclaimed to God?” Tasimanes using the same interpreter said that he would explain: “We asked for forgiveness from God for the deceased, for his own sins committed in his soul.” Retorting myself I said, “Very well, but the judge is merciful, indeed, and dispenses mercy; and he who will come as judge of every race of men, even according to you, is Christ. You must be addressing, therefore, the prayers and the exclamations to Him. Thus you, too, invoke him as God, as we do, who believe that as an inborn Word of His he is indivisible from the Father; for there was no time when God was without reason or without the natural word.” (Source: Saint Gregory Palamas, Littera & Dialexis Patrologia Migne –PG, CL, COL. DCCCVIII)

The orthodox have to deal with their patriarch admitting that non-Christians worship the same God as Christians:

• "§58 The story of other non-Orthodox Christian religious traditions is not yet finished, and Orthodoxy affirms that like other non-Orthodox Christian bodies they only find their coherence and clarity within the Orthodox Church. As for other religions, the Orthodox Church takes encouragement from the words of the Apostle Paul to the Athenians at the Areopagus: “What therefore you worship as unknown, this I proclaim to you” (Acts 17:23). From this the Church is given license to proclaim that the true God in whom all humanity lives and moves and has its being is worshipped by peoples everywhere, Christian and non-Christian alike. And this makes her only more eager to make all persons and peoples aware that the face of this one true God shines forth unobscured in the face of Jesus Christ. Moreover, the Church—illumined by that radiance—enters into dialogue with other faiths fully prepared to be instructed by many of their own speculative, cultural, and spiritual achievements. It may be that, just as the Church of the early centuries profited from and in time baptized many of the philosophical, religious, and cultural riches of pre-Christian Europe, Asia Minor, and the Near East, so too may it now discover new ways of articulating the deposit of faith or new ways of thinking about its cultural expressions and conceptual forms by exposure to, say, the great philosophies and faiths OF INDIA, or to the traditions OF CHINA and the greater FAR EAST, or to the spiritual experiences of tribal peoples throughout the world, and so on. Again, as Justin Martyr insisted, whatsoever is true and godly is welcome to us, for the Logos is everywhere and shines forth in all places" (citation: For the Life of the World: Toward A Social Ethos of the Orthodox Church, published by Holy Cross Orthodox Press in 2020. The document can be accessed at the Greek Orthodox Archdiocese of America website: Social Ethos Document)

• Note that the Ecumenical Patriarchate of the eastern orthodox was one of the founding members of the World Council of Churches