From a Christian perspective, when Islam first came on the scene it was considered a Christian heresy. I know St. John of Damascus deemed Islam to essentially be the Arian heresy. It is more than obvious from looking at a lot of Muslim practices that they had contact with Christians and their practices at the time.
I find that the answer of whether or not we all pray to the same God changes (among Muslims) depending on the conversion tactics being used with particular groups of people. If someone is attempting to convert the New Age all paths lead to God people, then everyone worships the same God. If someone is attempting to show a dedicated Christian that the Trinity is a wrong belief- then we suddenly become polytheists and certainly don’t worship the same God.
Anyway, I’m not an expert on Islam, but I have read the Quran (I once found a website that listed three English translations side by side and took advantage of it) and I’ve read quite a few hadiths. There is one problem I’ve seen (more educated folks than me discovered it) regarding God as presented in the Quran.
Muslims say that Jesus is the Word of God and Spirit. It says this in the Quran regarding Jesus.
Kalima tullah- "The Word of Allah (God)
Ruhun min Allah - “A spirit from Allah (God).”
If so, and to all evidence the Quran does seem to claim this about Jesus, and since they believe that Jesus isn’t God, then they have a serious problem. If Christ isn’t God, then they believe that God is without Word or Spirit.
In light of this, I personally say that Muslims don’t worship the same God as Christians. Muhammad took the things he saw Christians doing and made up his own religion without having anywhere near a complete understanding of the implications of certain things. Another example: Even Muslims believe that Jesus was born of a virgin. Muhammad certainly wasn’t. They have no concept of why that is an important thing and the implication of it.
We do not consider modalist Pentecostals to be Christian as they also deny the Trinity. We do not consider the LDS faith Christian as they have a different concept of God. We don’t consider the JWs Christian as they consider St. Michael to be Jesus. These groups fall very short and deny Orthodox Christian teachings on the Godhead. These things matter.