841 The Church's relationship with the Muslims. "The plan of salvation also includes those who acknowledge the Creator, in the first place amongst whom are the Muslims; these profess to hold the faith of Abraham, and together with us they adore the one, merciful God, mankind's judge on the last day."
Near as I can tell, the mohammedans assert that God is One, that He is merciful, and that He will judge mankind.
If so, they are correct in so asserting. That may well be the extent of their correctness ... and what I have seen suggests that they go off the rails pretty quickly after that. Still ... it puts them ahead of Marxist Communists, for example, who deny God entirely.
While not the only defense of my statement: "The Roman church claims we (Christians v. Islam) serve the same God," your evidence as per your last comment, serves the purpose for now. To accept the claim by the Muslims that they in fact serve the God of Abraham - Namely YHWH - One must needfully accept it as such by face value alone, without consideration of the evidence. IOW, "They do because they SAY they do."
Mohammed (to include later agents acting in discipleship) declares himself to be a prophet of a god. In order for that god to be YHWH (according to YHWH), there must needfully be 100% consistency with what came before.
This is an incredibly important point of order which is ignored by Christendom (not just y'all). YHWH's evidence that He (and He alone) is GOD is that He declares the end from the beginning - That what He said in the beginning will in fact be the result in the end. Without that proof, there is no evidence that He is in fact God, and worthy of our praise and worship. Therefore, that which was declared before must needfully be maintained, and is the primary reason for the general statement by YHWH: "I am YHWH and I do not change".
This sets Him apart particularly in the field of 'gods' - Every other religion/deity that I have examined allows change in what 'the deity said' to occur according to some kind of tradition or primary authority of the priest/king/etc. This is, as an aside, the primary reason I am opposed to y'all - Your tradition causes the Word of YHWH to be made null.
In that, those who lean upon the original covenant(s) (Torah primarily, entirety of the Tanakh or OT included) to add additional volumes to His Word (LDS, Muslims, and I would include even the New Covenant if not taken in the light of it's originating documents) and use those volumes to change what He said in the beginning can only be nefarious and profane in origination... a 'god' co-opting the truth of YHWH. While many may follow this prophet and his 'god' in ignorance, the fact remains that they are not following YHWH, unless they follow the One who does not change YHWH's words. That is what it really is all about at the root.
At that point, one must examine the deity professed and declared in those later additions to see if it is indeed, YHWH, and/or of YHWH.
I can make a defense of Yeshua as Messiah according to the Old Covenant... and what He (and His agents) says in the New Covenant, if taken in a context relying upon the Old Covenant, does not break the Law or the Prophets. Therefore I can declare Yeshua as Messiah, and I will follow Him technically as the Great Teacher and Prophet (technically = pertaining to the letter of His words, following His example as a disciple thereof)...
I cannot do so for the Koran, or it's prophet... and I will lay odds that you will not be able to either.
Allah is not YHWH.
So in this instance at least, The Roman church is dreadfully mistaken.