Yes, it is Catholic doctrine that we all worship the same God. If Cardinal Burke does not adhere to it, he is on this issue outside Catholic doctrine. As you say, it is not a place where legitimate opinions may differ.
I think Burke’s perspective can possibly defended on these grounds:
The Council Fathers did not vote on the various translations of the Vatican 2 documents, only the Latin original. (Similarly, the official text of the Catechism is the Latin, and all translations are supposed to be corrected, when necessary, according to that text.) Latin does not have a definite article similar to our word “the.” With this in mind, I think one could legitimately defend the position that the Catechism and the Vatican 2 documents only say that Muslims “worship one God,” not necessarily that they worship the same God we do.
On this note, I think it is worthwhile to show that the alternative translation “worship one God” is actually Also used by papal sources. For example, consider the way the papal book Crossing the Threshold of Hope (by St. John Paul 2) quotes the Vatican 2 document Nostra Aetate Paragraph 3: “The Church also has a high regard for the Muslims, who worship one God, living and subsistent, merciful and omnipotent, the Creator of heaven and earth.”
source This contrasts with the typical English translation of Nostra Aetate Paragraph 3, which says, “The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth.”
source
From this, it appears that it is possible to defend an alternate translation of the phrase “They worship the one God” so that it only says “They worship one God,” not necessarily the same one we worship. If this position is an option, which is a position that I think is defensible, then it seems to remain open to interpretation whether Muslims and Christians worship the same God or not.
To be clear, my own position is that they do worship the same God we do and that this is the classic and traditional understanding. It goes further back than Pope St. Gregory VII (from the 1000s). But I do think you can remain a good Catholic and disagree with this position, if you safeguard the infallibility of Vatican 2 by maintaining that (for various reasons) it does not actually define that we all worship the same God. I think that is a defensible position, but, in my opinion, the position that we Do worship the same God has the stronger evidence from Tradition.
Actually I wonder if the two positions are, in some sense, talking past each other. Perhaps there is One Sense in which we Do worship the same God and one sense in which we don’t. C.S. Lewis’s book “The Last Battle” argues that we do not worship the same god as non-Christians do but that all their sincere prayers end up going to the true God anyway, because nothing good can go to false gods and nothing bad can go to God. (
source) This seems to Result in a similar doctrine as what we hold, while approaching it from the perspective that we do Not worship the same God. That makes me think we might Really mean the same thing, even though we are using different words.
I hope that helps. God bless!