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Muslims are in no way worshipping the same God.

I might agree, but the Catechism doesn't. As posted earlier on the thread:

"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."

Note that the Catechism says that Muslims "adore" God. What does "adore" mean? To clear things up vis a vie the language used in Catechism, Cardinal Antonio Bacci states that adoration is the worship of God:

....The worship of the Saints is an act of veneration (dulia), not of adoration (latria), which can be given only to God. It is wrong to imagine, as many Protestants do, that by praying to and venerating the Saints we subtract something from the homage we owe to God. The veneration of the Saints and the adoration of God are entirely distinct activities. Moreover, the Saints are the faithful servants of God and intercede with Him on our behalf. By venerating and invoking them, we honour the Giver of all holiness. If anyone, on the other hand, were to disregard the worship of God in favour of devotion to the Saints, he would be making a serious mistake. A person who goes into a church and rushes over to a statue of the Blessed Virgin or of one of the Saints, without giving a thought to the living and real presence of Jesus in the Blessed Eucharist, is developing a false and sentimental piety.

And the Catholic Dictionary New Advent agrees with the Cardinal, in their definition of "dulia"

...a theological term signifying the honour paid to the saints, while latria means worship given to God alone, and hyperdulia the veneration offered to the Blessed Virgin Mary....A further distinction is made between dulia in the absolute sense, the honour paid to persons, and dulia in the relative sense, the honour paid to inanimate objects, such as images and relics.