The short answer is that it depends on what you mean by “same” and “God” in this case. I’ve seen cases for “no” and for “yes” posted by smarter folks than me since a Wheaton professor got in trouble for implying this was the case. Much of this seems to turn on the understanding of both faiths as being part of the Abrahamic tradition that also includes Judaism. The history of both religions is certainly intertwined and even the similar linguistic roots of “Elohim” and “Allah” are not that difficult to discern.
Let’s put aside all that, for the sake of argument let’s say, no, the monotheistic tradition of Abrahamic religion doesn’t count for anything. There is still a reason to hesitate in saying that Muslims worship a different god. One of the principal New Atheist arguments is that there have been thousands of deities worshipped by humans in recorded history, and most of us don’t believe in most of those gods. The idea is that almost everyone today is an atheist with regard to Odin, Zeus, Osiris, Chemosh, Baal, Ishtar and so on, that the difference between an atheist and (mono)theist is that the atheist disbelieves only one more god than the theist.
One of the better responses is not that everyone doesn’t believe in everyone else’s gods, but rather that the overwhelming majority of humans through history have had a conception of the divine, and that we are not atheists about other gods per se even as reject other conceptions of god(s). This is not to say that we are all universalists or that our distinctions don’t matter – it should be readily apparent today that they do. Rather it means that we disagree sharply about who or what is divine.
Do I have the same conception of god as a Muslim? No. We disagree about who God is and what God wants from us and how we can even know or approach God. I am not a Muslim, I am not going to give up bacon or beer for starters because I do not perceive any injunction against them. But there is still something common in our attempts to approach God, in spite of all our obvious and real differences.