My dear brother Devout :)
People understand this teaching in different ways, and I'm not prepared to say which is the correct one. However the Vatican II documents Nostra Aetate and Lumen Gentium, as well as the Catechism, both mention Islam and Catholics are certainly bound by those teachings - ie we must obey them, give assent. However people debate precisely what the Church is actually telling us.
Personally, I believe that Muslims do indeed worship the same God but simply have a different, that is inferior, understanding of Him (not that anyone can understand Him in Essence - as he is in Himself). Indeed we are all united, all of us whether Christian or Muslim or else, in not being able to fully comprehend God who is so very far above our finite understanding. We thus have no cause, despite the Truth we have in Christ, to feel ourselves superior to our Muslim brothers and sisters. We are all united in the search for truth.
Without Christ, and without the direct divine revelation of Judaism, Islam - despite much truth and inspiration from the Holy Spirit - has a deficient understanding of the One God - which is understandable given that without Our Lord he is so cold and distant and "other" and unknowable. In this respect, and influenced by the harsh Arabic desert environment from which Islam emerged, the Muslim conception of God can often appear to us Christians to be rather violent and xenophobic. This is of course not universally true, for there are stunningly beautiful, inspired ayats in the Qur'an often mixed in with this cruelty - for example the very moving one which says that if we take just one human life it is accounted as if we have slain the swhole of mankind; or the one which says, "you have your religion and I have mine".
Sufi Islam, inspired perhaps by the ayat of the Qur'an which said that God is closer to us than our jugular vein, took great strides towards creating a much more personal, intimate understanding of God that is very close to Christianity.
I wanted to add three quotes from three holy Popes - a Blessed, a Saint and a Servant of God - two from recent times and one from a thousand years ago on Islam. Pope John Paul II's statement below is illuminating (in fact I encourage you to do a google search and read the full address):
"...Christians and Muslims, we have many things in common, as believers and as human beings. We live in the same world, marked by many signs of hope, but also by multiple signs of anguish. For us, Abraham is a very model of faith in God, of submission to his will and of confidence in his goodness. We believe in the same God, the one God, the living God, the God who created the world and brings his creatures to their perfection...The Catholic Church regards with respect and recognizes the equality of your religious progress, the richness of your spiritual tradition...On this path, you are assured, of the esteem and the collaboration of your Catholic brothers and sisters whom I represent among you this evening..."
- Blessed Pope John Paul II: Address to young Muslims in Casablanca, 1985
"...This good action was inspired in your heart by God, the Creator of all things, without whom we can neither do nor think any good thing. He who enlightens all men coming into this world (John 1.9) has enlightened your mind for this purpose. Almighty God, who wishes that all should be saved and none lost, approves nothing in so much as that after loving Him one should love his fellow man, and that one should not do to others, what one does not want done to oneself. This affection we and you owe to each other in a more peculiar way than to people of other races **because we worship and confess the same God **though in diverse forms and daily praise and adore Him as the creator and ruler of this world. For, in the words of the Apostle, 'He is our peace who hath made both one.' This good action was inspired in your heart by God....This grace granted to you by God is admired and praised by many of the Roman nobility who have learned from us of your benevolence and high qualities . . .] For God knows that we love you purely for His honour and that we desire your salvation and glory, both in this life and in the life to come. And we pray in our hearts and with our lips that God may lead you to the abode of happiness, to the bosom of the holy patriarch Abraham, after long years of life here on earth..."
*- Pope St. Gregory VII, Letter XXI to Al-Nasir the Muslim Ruler of Bijaya (Algeria), 1076 *
Of the above Pope John Paul II said in 1990:
"...I close my greeting to you with the words of one of my predecessors, Pope Gregory VII who in 1076 wrote to Al-Nasir, the Muslim Ruler of Bijaya, present day Algeria...These words, written almost a thousand years ago, express my feelings to you today as you celebrate ‘Id al-Fitr, the Feast of the Breaking of the Fast. May the Most High God fill us with all His merciful love and peace..."
- Blessed Pope John Paul II, Message to the faithful of Islam at the end of the month of Ramadan, April 3, 1991
"...Now [we refer] to the adorers of God according to the conception of monotheism, the Muslim religion especially, deserving of our admiration for all that is true and good in their worship of God..."
- Servant of God Pope Paul VI, Ecclesiam Suam 107, August 6, 1964