All right; I know this one has been posed before,

However, the threads I've found on the subject tend to drift into treating the question as a religious posit.
Being an agnostic, and considering, personally, Abrahamic mythology as being no more or less likely than any other, I pose the question purely as one of sociology and identity (the reason I've posted it here, and not in the ASB forum).
Assuming no biblical Ishmale (i.e., the biblical story of Abraham tells only of Isaac), what are the effects on the development of Arabic racial and cultural identity? Is Islam altered to become something unrecognisable, or butterflied as a result of cultural divergence? Assuming a recognisable Islam, is there a fundamental difference in its relationship towards the Jewish people as a race, and Jews in general? Is something apparently so insignificant likely to cause such massive butterflies as to make the modern world (or at least, the Middle-East and its political and cultural landscape) vastly different from OTL?
This assumes a POD about the sixth century BCE or earlier, so, in principle, nay, nothing like recognizable Islam exists, and even Christianity is theoretically a very though sell.
However, let's assume a Butterfly Killing approach where only the effects you can precisely derive from the POD are taken into account, while the rest stay EXACTLY the same.
This is not realistic, but whatver.
The Bible is slightly different from the start. What changes? Hard to say, too little evidence. Let's say nothing.
The story about Ishmael is Jewish, and attests a feeling of kinship among Jews of the time (Seventh or Sixth c. BCE probably) and those "Arabs" who identified some Ishmael, or Ismail (whatever) as eponymous ancestor. Jews and various Arabian communities were trading partners, political allies or enemies, co-subjects of various Imperial realities. At some point they developed mythical narratives of common ancestors (which may even include a grain of truth, for all we know, but that is not the point). We may assume that this, in turn, fostered relations.
In later times, Jewish communities established in Arabia IOTL. Nabatean and Jews has extensive contacts, although not always friendly ones. Would that occur to the same extent without a prior kinship narrative? And if so, wouldn't it spur the birth of another kinship narrative? Mayhaps referred to somebody more obscure, farther removed up the genealogical tree.
Who knows? But if the general outline of Jewish-"Arab" relations is the same, similar narratives arise. If not, the POD is not the lack of a story about Ishmael, is the lack of the sort of interrelation between Israel and Ishmael that led to the creation of the story. Which, if continued, would indeed change the world deeply on the long term, with the possibility of a non-monotheist, or very differently monotheist, Arabia down the line.
But going for minimal changes, letting the story not to be there but the relations still existing mostly unchanged. Let's go for we can only calculate, that is, only the little sources say. Discount the thousands of invisible threads, let's say that there is no direct causal relation we can tell between the story of Ishmael and the birth of Muhammad, his receiving a revelation. Let's assume that the genesis of the Qur'an and Sunna is historical, not divine. What we can calculate, is that there would be no reference to Ishmael in *Islam, but *Islam would still be the instrument for Arabs to claim their place in the montheist world. They've been exposed to the same religious environment, after all. They would still find plenty of biblical material suggesting relationships with the Jewish world, the Christian world. *Islam would elaborate on that. They would construct narrative of Moses coming to Mecca maybe (Iethro may be easily construed as ancestral to Arabs).
Ludicrously unlikely of course.