Do Muslims Worship the Same God as Christians?
Either there is one God or there isn't. If there is one God, then let's stop talking about gods, except in reference to pagans. Pagans believe in gods. All monotheists believe in one God, by definition. There is one God and we monotheists believe in one God, therefore we all are trying to obtain Grace from the same God. It's simple logic.
I will freely admit that most monotheistic religions have a pretty sketchy theology. Many pastors are pulling their sermons out of their asses, but if they are "men of good will", then they are in pursuit of Logos -- no matter how foolish their methods.
Islam is just a name for another category of dumbed-down Protestant denominations, all filled with potential saints attempting to conform themselves to a meager slice of the Faith. There are millions of Protestants who believe all the important tenets of Islam. In fact, there is evidence that "Muhammad" (Messenger of God) is just an Arabic translation of "Christ".
If I grew up in Iran absolutely cut off from Christianity, then I would be a good Muslim in pursuit of Logos. If I was a Southern Baptist taught from birth that the Romanish Church is the whore of Babylon, then I would reject Her out of hand and never investigate Her and I would be the best Southern Baptist I could be, in pursuit of Logos. When Moses died in Moab 1200 years before the Incarnation, he died in pursuit of Logos.
The concept of Logos concentrates upon our points of agreement and avoids our disagreements. We may disagree on important theological points that must be hammered out at some point, but until then let us come together as disciples of Logos. Then when we do begin to discuss divisive issues it will be as friends, not enemies.
He is risen for one and all. Islam has no understanding of Logos. They have no writings about Logos. However, individual Muslims may or may not express an interest in Logos. That's a start.
All man-made faiths have some flaw, some element of untruth that the Catholic Faith does not contain. She is unstained even though Her priests, bishops and popes be sinners one and all. To be a sinner is to be separated from God, and therefore all of us in the Church Militant are guaranteed to be sinners. However, while ignorance is not a sin, willful maintenance of that ignorance is a sin. We all have the responsibilty to become better than we are, guided by the Holy Spirit.
I believe most followers of those lesser faiths "hold their noses" or "suspend their disbelief" when faced with these errors. They continue to be so attracted to the undeniable truths held within these faiths that they persevere in the face of these logical disconnects. If they became aware of the unstained truths held within the Magisterium they would run into the nearest Catholic Church and declare their undying fealty. Unfortunately the Catholics there would probably call the cops.
The Magisterium contains the "Fullness of Truth", which is not to say it is Absolute Truth, but it is the nearest we have come to Him in this world. For Truth is a Person, not a collection of facts.
Do we claim certain knowledge that Moses is in Hell? If we are not certain, then there is hope for "all people of good will", which might well include a Muslim or two, maybe from time to time a Hindu, and perhaps even an occasional Jew. It's possible, although modern Judaism has defined itself as rejection of Logos. Moses was not a member of the Catholic Church, but he might well be a member of the Church Triumphant.
Don't confuse pursuit of Gnostic "enlightenment" (generally referring to Lucifer, the 'light-bearer') with pursuit of Logos. They are entirely different things. According to Freemasons, if you don't pursue their version of "enlightenment" then it is because you were born profane. You don't have a chance, according to their religion, because you were born to be cannon fodder. In their minds you are either born a speed bump or you were created to be a useful tool to advance their own personal spirit quest, nothing more.
Christianity is quite different than that, although many self-styled Christians think more like Deists than Christians. Christ is present here, all around us, in every breath we take. We represent Him (or misrepresent Him) in every word we speak, in our every act. We know when we wrong Him, and we know when we do something that makes Him happy. We know this in everyday life, quite outside of Scriptural contexts.
Is it so impossible for us to imagine that non-Christians feel Him too? Christ has a special place tailor made in Heaven for each human being. We all get to choose where we end up. However, if we are kept ignorant of Christ this choice becomes difficult. The choice becomes even more difficult when we are raised in anti-Christian environment. But it is our choices that make us Christian, not our place of birth or language.
Our job as Christians is to save everyone. Not from discomfort, although we are urged to give comfort. Not from loneliness, although spending time with a lonely person is a great way to share Christ. Our number one task is to inform our brothers and sisters about the truth of Christ. Doing this requires constant prayer, so that might be considered the foundation or the inspiration to persevere in our task; but don't forget what is Job One for us.
It is possible to sense what is right, and turn away from what is wrong, without ever being exposed to Christian history or dogma. But we must not leave this momentous decision (or series of decisions) to luck or chance. We must not force the Holy Spirit to do what we ourselves should be doing. We must do our best to show Christ in everything we do and say. We must be His hands and feet on this world until He returns. We must speak to our brothers and sisters about Christ, so that they might choose rightly.
This is not always easy. Christianity is not always easy. But I am not looking for the easy way out. I am not interested in "what are the minimum requirements to get into Heaven". I am afraid to ask for the maximum, so I am working on something in between. I am deliberately working out my salvation "in fear and trembling". Don't be so confident that you have what it takes to gain heaven, and that some Muslim dude is forever barred. You might just be wrong.
Short answer: Not the same.
Things with different definitions, even slightly different definitions, are clearly not the same thing.
(See Catholic dogma for details.)