Worship the same God?

The anniversary of 9/11, Koran burning, and the beginning of Rosh Hashanah has created many opportunities for discussion on a few of the Catholic web logs I read. In a number of those discussions I see it mentioned that "We all worship the same God." The "we" refers to Muslims, Jews, and in some discussion Protestants.

Worship in its most general sense is homage paid to a person or a thing. Christian worship is "homage paid to God, to Jesus Christ, to His saints, to the beings or even to the objects which have a special relation to God."

It could be argued at length whether another religion/faith or even a specific person is or is not actually worshiping God. One alternative possibility is that the other religion/faith practices Idolatry. Idolatry is divine worship given to anyone or anything but the true God. Another possibility is heresy which is defined by St. Thomas as "a species of infidelity in men who, having professed the faith of Christ, corrupt its dogmas." St. Thomas goes on to say:

The right Christian faith consists in giving one's voluntary assent to Christ in all that truly belongs to His teaching. There are, therefore, two ways of deviating from Christianity: the one by refusing to believe in Christ Himself, which is the way of infidelity, common to Pagans and Jews; the other by restricting belief to certain points of Christ's doctrine selected and fashioned at pleasure, which is the way of heretics

While idolatry is certainly not worship of God, Those who are of heresy and infidelity practice, at best, an imperfect worship of God.

Discussion could take place about the various degrees of worship, and when Infidelity or Heresy can turn to Idolatry. But let us put that all aside for a moment and ask ourselves a question. What does it mean if the statement is true that "we" worship the same God? Does it mean that those other religions/faiths are not called to conversion? Does it mean that we as Catholics should not pray and work towards the conversion of their souls? Is it a means of just recognizing "common ground"? For what purpose is the statement made?

On the question of "common ground"... it seems that when attempting dialog with a Muslim or Jew (and sometimes the Protestant)... simply saying that we worship the same God is not received well. It is not received well, because often those other religions/faiths do not themselves believe that we worship the same God.

I leave things open ended a bit in hopes to encourage discussion of the topic. Please remember to keep it Charitable.