Apparently, this is an issue for some Christians. The Catholic Church has definitively declared that, yes, God and Allah are the same, and Muslims and Christians worship the same God.
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.Catechism of the Catholic Church, 841. Cf. Vatican II, Lumen Gentium, 16
For some Christians, this is an issue because Muslims deny that Jesus is the Son of God, whereas Christians claim that Jesus is the Eternal Son of God, the Second Person of the Holy Trinity. Muslims deny that God has, or could have, a Son. So if Christians worship a Trinity of Persons in God, and indeed worship Jesus as God, and Muslims deny any plurality of God, then they seem to be talking about two different things.
I think the best way to approach this is to distinguish between reference and description. (A more complete treatment can be found at “This Everyone Calls God“.) Names can have the same referent, that is, the same thing they point to, but point to it under different descriptions, or lists of attributes. There are some classic examples in philosophy. You can talk about Socrates as being the teacher of Plato or the husband of Xantippe. “Teacher of Plato” and “husband of Xanthippe” are two different descriptions, but they both refer to the same person who goes by the name, Socrates. Or the evening star, called Hesperus, and Phosphorus, the Morningstar. To the ancient Greeks, they thought these were two different stars, but it was later discovered that they were, in fact, the planet Venus. So there are two different descriptions, one as appearing before sunrise, and the other as appearing soon after sunset. But the two descriptions have one and the same referent.
From this perspective, the names, “Holy Trinity,” or “the Father of Jesus,” or “the God of Abraham, Isaac and Jacob,” and “Allah” can all have the same referent, i.e., the same thing that is referred to — the same creator of the universe — but they have different descriptions, or lists of attributes. These terms have the same Divine Being as referent, but under different descriptions. And obviously “the God of Abraham, Isaac, and Jacob,” that is “Yahweh,” the God of the Jews is not (or is not understood or described as) a Trinity of Persons (even if the reality is that He is). But Jesus explicitly says that his Father is the God of Abraham, Isaac and Jacob.
Jesus Himself confirms that a person can worship the true God, but under a different description. When Jesus is talking to the Samaritan woman at the well, he says
You people worship what you do not understand; we worship what we understand, because salvation is from the Jews.John 4:22
Thus, Jesus indicates that people can worship the One True God, without understanding Him completely, and indeed, without understanding that Jesus Himself is that One True God.
Saint Paul implies the same thing when he is preaching to the Greeks in Athens at the Areopagus. He points out that they have a temple to an unknown God, and then he informs them that this unknown God is in fact, Jesus who was raised from the dead.
What therefore you unknowingly worship, I proclaim to you.Acts 17:23
Even Saint Paul acknowledges that the Greeks were worshiping God before they knew who God was, or that God became incarnate in Jesus. So Scripture itself indicates that it’s possible to refer to the same one God, but worship him or describe him differently and under a different description.
This seems to be an issue for some Christians because if you say that Muslims, for instance, worship the same God that Christians do, somehow you’re equating Islam with Christianity or the Catholic faith. And these statements are not saying the same thing. They’re two different questions. Who Muslims worship is one thing, and it just seems to be the case that they’re worshiping the One True God. Whether they are worshipping him correctly or as he has revealed himself, or whether they understand all truth about Him? — obviously not. From the perspective of the Catholic faith, they miss out on a lot. But that doesn’t mean they’re referring to a different thing.
And so simply to say that the One True God is being referred to by both Muslims and Christians (and Jews or other monotheists) does not mean that the different religions are equal, or there’s no difference between different religions.
It is yet another question whether Muslims or other non-believers can be saved and can go to heaven.