No such scriptual teaching exists, and "by definition" only asserts a postulate (which, in this case, is a postulate that is disproven when the argument takes full form). Jesus was a Jew, so it is logically consistent to assert that the God of Christianity is the God of Abraham. No such intrinsic link exists in the case of Islam. Since "Allah" was a name given to a deity of the Kaaba pantheon (one of 360 gods worshipped at that site before Mohammed's band of thugs destroyed the others), there is no historical or logical link to assume Islam is merely a change in form of worship of the Judeo-Christian Deity when instead it is far more likely to be a revision of worship of the deity who instead already had that name at that location. Other aspects of Islam (the black rock, the Hajj, etc.) which were part of pagan Allah worship only reinforce that conclusion. The idea that the Kaaba was built by Abraham, or that the "Allah" worshipped there was the god of Abraham, is Islamic revisionist history easily disproven by the archaeological record.
If historical facts were insufficient reason to reject such an argument, then the natures of the religions themselves can be considered. Islam (Sura 112) asserts that Allah begets not, nor is he begotten. The God of Christianity, on the other hand, has a begotten Son named Jesus. If I were to say I know George Bush, and you say you do too and we begin talking about him, and you mention his two daughters while I start talking about his lifetime vow of chastity and the fact that he has no children, then it becomes clear that either one of us is lying about George or that we are talking about two different George Bushes.
The philosophy of the world asserts that all religions are of equal merit and validity. But if you put your worldly philosophy and "feelings" aside and use the reasoning powers that God gave you, you can easily see that such assertions are self-evident falsehoods. From a logical or historical perspective, even the Zoroastrians have a more valid claim on the God of Abraham than the Mohammedans do.
This is a historically fallacious argument.
The Jews hardly worshipped a new deity when they began worshipping El. The Cannanites and Ugarites worshipped an El who looked nothing like the El in the Bible. Yet that did not prevent Moses from using that name for God quite profusely.
The fact that pagans used the same name in worshipping one in their pantheon does not preclude Allah from being the same entity that the Jews and the Christians worship, albeit understood quite differently (and, as a Christian, I must say wrongly.) Arabic Christians refer to God as "Allah," and ultimately, "Allah" is a linguistic corruption of the tetragrammaton YHWH.
This debate - whether or not the entity is the same - is at best a red herring. What matters, rather, is how God is understood by the Jews, the Christians, and the Muslims. As a Christian, I have to say there is only one right way to understand God, and neither the Jews nor the Muslims fully comprehend it, though both may have an inferior understanding of the True Faith. What word they use is irrelevant. Islam and Judaism, by their terms, claim to worship the same God as I. I must take them at their word, and engage them on theology, not semantics.