Much of this is semantics. "Allah" is simply the Arabic word for "God". If you are an Arabic-speaking Christian, "Allah" is the word you would use for the Christian god.
While Christians and Muslims have drastically different conceptions of God, what does it mean to worship the same god or a different god? Christians believe that the trinitarian God is a reality, and that the god described in the Islamic texts doesn't exist. If Christianity is true, either the Muslims worship the same God, but have a misconception of His character and nature, or they worship a non-existent entity (in which case they are worshiping nothing). They can't worship a different God unless there exists more than one God. And if both "Yahweh" and "Allah" exist and are different gods, then Christianity is not true, since Christianity teaches that there is but one God (and Islam would not be true since Islam also teaches there is but one God).