Do Muslims Worship Our God?

I was recently reading an online argument that went something like this:

Catholic A: Muslims worship the same God as Christians.
Catholic B: No, they don't, they worship a false God.
A: Yes, they do. The Catechism says, "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.[841]"
B: Yeah, I know, but that's not an infallible pronouncement, it's just "official teaching."
A: Thanks, but I'll trust the Catechism before I trust "Catholic B" from some online forum.

And so on, and so on...

I think there's a little room for a middle ground on this one. Let me explain:

Let's say you and your friend both were hired by this guy named Bob. Then you were sent to different parts of the country to do work for Bob's company. You both were to send letters each day, to the same address, to let him know how you were doing and if there was anything you needed. Bob himself would read the letters, and then you would receive letters in response from Dave, and your friend would receive letters from Fred, both supposedly being dictated to by Bob.

After some time, you compared the letters you were receiving with those your friend received, and it turned out that you were receiving some of the same information, but sometimes the information would be contradictory, as if you were being instructed by two different Bobs. Apparently, either Fred or Dave was not sending true correspondence from Bob.

The point I'm trying to make is that there is only one God, and we both know him to a degree, because we both have history with him at some point. But the teachings of Mohammad contradict the Catholic faith in many areas, so they must not be true in their entirety. What Islam has preserved intact from Christianity and Judaism is, however, true.

As I see it, this means that we do both pray to the same God, because if we address our prayers to "the one God," then he receives them. Also, to the extent that they recognize true attributes of God, they adore the same God. And finally, to the extent that they recognize the teachings and commands of God, they serve the same God.

So, to sum it up: Do they pray to the same God? Yes. Do they adore the same God? Partly. Do they serve the same God? To an extent. Do Muslims believe all we believe about God? No.

I think the Catechism attests to the idea that Muslims may love [adore] the same God as we do, but often serve a false god, when it says:

"In their religious behavior, however, men also display the limits and errors that disfigure the image of God in them: Very often, deceived by the Evil One, men have become vain in their reasonings, and have exchanged the truth of God for a lie, and served the creature rather than the Creator.[844]"

Another point, somewhere in there, is that God, being an actual being, and not just a philosophical concept, can be addressed or even loved without being understood. Thus, Muslims can love God and pray to God while not actually knowing him as well as Christians.

This is similar to how a child can pray to God without understanding him like an adult, or how an adult can love him without understanding him as well as the apostles understood him.