The subject of the religion of a politician is always tricky. One viewpoint is that it is personal and therefore none of our business. This ignores the fact that the behavior of people and their policies often has a direct connect to their religion, and in a Democracy it is our business to choose politicians based on how we think they will behave. Another viewpoint is that someone's religion is inherently unknowable, so why bother? Then the cynics chime in and note that the only deity that a politician cares anything about is himself. Another rhetorical angle is the "who are you to question his faith"? Technically, I believe we all have a faith and I don't question that. I just want to know "faith in What?" Is it a faith in the Flying Spaghetti Monster that drives atheists? What is it? From my perspective, a politician's religion does matter and we can list possibilities and impossibilities regarding the First Religion, but we must acknowledge that only at the final judgment will the nuances be made clear.
First we must consider the American context. For the last century, if anything correct was taught about Christianity (or Islam) in the government schools, clerics would immediately start litigation alleging a violation of "the Constitutional separation of Church and State". If anything wrong was taught, this would be defended by the same clerics as "freedom of intellectual inquiry" and that to become mature thinkers, students would need to be exposed to ideas they didn't like. The result being that most graduates from Ivy League schools these days have notions of religion, history and philosophy that are completely garbled and divorced from reality. To some extent both Obama and I are products of this same indoctrination program, as are just about all Americans. To find out more about Christianity, we are typically shaped by the denomination we associate with - if indeed we do associate with one - or else we spend a lot of time investigating matters ourselves, which few do. Then there is the Bible itself, which is the primary source of Christian teaching, unless you are a Papist or some other group that deprecates the Bible.
So one way to answer this question is to look at the religious organizations that people associate with. Obama was famously with the United Church of Christ. This is an extremist sect that uses the Bible, but negates most of the teaching. For example, it was at the forefront of ordaining gay priests and promoting hyper-promiscuity as if the Biblical command to "love your neighbor as yourself" was primarily to be understood in sexual terms, in spite of all the more numerous commands to restrain your sexual impulses. I listened to an Easter sermon from Obama's church (a Black congregation) which featured imagery of Christ and the Crucifixion, but then switched to imagery of a lynching, ignored the resurrection, and offered up a conclusion that was purely Victim Theology: The belief that others have sinned against me, which is the only thing that matters. Certainly the Black community has suffered, but the UCC has not only been useless in helping the Black family to recover, they have been actively working to make the problem worse by transferring the ownership of the concept of the family to the LGBTQ activists.
Then there was Obama's preacher with his famous "God Damn America" sermon, which has a similar ring to "Death to America". One is tempted to think that the Mohammedans merely wish for America to die, but Jeremiah Wright wants America to burn in Hell forever. This certainly reflects the view of a large portion of America. The Bible teaches us to look at ourselves for the answer, but Victim Theology teaches us to look at others and assume that what ever they have, they got by stealing. The result is that many who are richer and better off than 90% of the world's population are burning with rage about how they have been mistreated. Another result is the White House's foray into the War on Cops.
The most curious Christian variant is that of the Social Justice movement. Orthodox Christians are commanded to do good and charity with their own time and money. Leftists have famously improved on this by doing charity with other people's time and money. Just as taking out loans and investing in the stock market permits people to make a huge profit compared to the dumb heads who only invest what they can spare, so a penniless deadbeat can do charity that vastly exceeds what any Orthodox Christian can do. The deadbeat along with the Pope can look down their noses at the churlish Christians who have constrained themselves by their depravity to never give more than 100% of what they have.
So where was I? Obama's religion. I don't see any evidence that his religion has any connection to Islam. Clearly it is informed by Christianity and the Bible in some sense, but we must keep in mind that even Satan quoted scripture. I think Obama's religion is simply a reflection of what is preached in the government schools by America's intellectual elite along with many of the modernist churches, which can be characterized as follows:
"But false prophets also arose among the people, just as there will be false teachers among you, who will secretly bring in destructive heresies, even denying the Master who bought them, bringing upon themselves swift destruction. And many will follow their sensuality, and because of them the way of truth will be blasphemed. And in their greed they will exploit you with false words. Their condemnation from long ago is not idle, and their destruction is not asleep." - 2 Peter 2:1-3