Im not sure that is possible if you look at it from the perspective of someone within each of those faiths, otherwise Christians would have to follow islam (if they recognised the god allah as being their own) but they cannot because the god allah begot no son - a fundamental and 3ssentially a pillar of the Christian faith. Therefore to a Christian allah can not be their god. It is a fictitious creation creation or a false god to them. That is the strict modern interpretation of islam. It has to be.
Superficially they seem to be the same god but paradoxically cannot be the same according to the respective faiths and their beliefs.
They simply hold different interpretations of doctrine and scripture, they all recognise that the God they believe in is the same...they do not believe the message from that a God is the same. Your reasoning, for what is, is simply flawed and they follow their beliefs of what God wants, not that God is not the same one. Pretty much everyone (with the exception of the Catholics who phrase the same theological understanding slightly differently) Jew, Christian and Muslim recognise the terminology "People of The Book" which means they all believe in the same God...Catholics say "People of the Word" which is essentially the same thing. Each religion recognises that each other has its basis on the scriptures, like Jews do not recognise the validity of the New Testament, they still recognise that Christians worship the same God who gave Moses the Law, equally Christians do not recognise the validity of The Quran, but they still recognise that Muslims worship the same God who gave Moses the Law, and equally the Quran itself recognises that Allah is the same God as worshipped by Jews, Sabians and Christians. Catholics recognise that all three religions derive from the Word of God, but differ on their interpretation of doctrine.
So no, a Christian doesn't have to follow Islam and vice versa...it simply means that they see the same a God, but a different Message.
Well Catholics don't take that view(so my mate says)
As I said Catholics have a slightly different t perspective based on their doctrine/dogma system...but they still recognise that God is the same, The Word is different. They do not recognise the association as People of the Book, they are People of the 'living Word of God", but the God remains the same as the God who gave the Law to Moses...ergo The same God.
I'm afraid that doesn't support your argument, Mohler is mainly talking about the Trinitarian interpretation of scripture being the only valid kind and making the leap that means that Muslims do not worship the same God as Abraham, but by that same argument you could say that all non Trinitarian Christians also do not worship the God of Abraham...in fact as Abraham worships the God of the Israelites, Moher is effectively saying that his God isn't the God of Abraham...which just goes to illustrate you shouldn't rely on the interpretations of American Evangelists particularly Exclusive Calvinists like Mohler, when it comes to exegesis. For example Mohler also believes that everyone not a Conservative Evangelical Baptist Calvinist like himself is a product of Satan, including all other Christians, let alone Muslims and Jews.
In any case they are really just discussion the use of Allah as the name of God...not whether they all worship the God of Abraham, as all three mainstream religions accept that they all do worship The God of Abraham.