So why are we told... The God of Abraham, Isaac, and Jacob? And, not the gods of Esau, nor Ishmael?
To generalize the term "God" is a misnomer, when this is done the noun "God" named is then only shown without a jussive (i.e. there is no expression of command) due to the fact that a generalization of character is completed.
Each "God" in each culture plays a role. In the Israelite Pantheon Yahweh is a storm God, thus he is equated as Ba'al in Canaan who is a storm God. Yahweh for example is only adopted into the Israelite pantheon, but doesn't originate with the Israelite's, he originates with the Canaanite's, as does El (pluarl Elohim), El Elyon. Granted these are all "God" nouns, but each one serves a function respective to their cultures.
Go look at Biblical Patriarchs which include Abraham, Isaac, Jacob, Esau, and Ishmael. By the time these texts are written you will need to examine the P, E, J, D sources where Biblical writ originates from. All 5 Bible characters are classified as Patriarchal, yet, we find distinct issues within the fray of each concerning a litany in early Rabbinic textual's and later Christian theology.
Since we can't prove that moʊzɪz (Moses) penned the Pentateuch there is a lot of disambiguation between early Judeo texts from about 1200 BC and on.
Bible for example contends that Esau is the progenitor of the Edomite's, however, the wife of Esau and an Edomite chief share the name Oholibamah, "tent of the high place." Also, without a theophoric element are the names of two metaphorical sisters in Ezekiel: Oholah "tent-woman" representing Samaria, and Oholibah "my tent is in her" representing Jerusalem. We find the names are metaphors for dwelling places, as the Israelite's used symbolism to explain their surroundings. Tents, were commonly shown as dwelling structures and as well had divinity values. Also, when the Israelite's separate from Canaan they set to distinguish themselves, Esau's wife is a Canaanite or I should say wives.
There is a lot of disdain for Esau as well. Gideon approached Succoth (to be identified with Tell Deir cAlla), and more to the south Penuel (most likely Tullul ed-Dahab) at the mouth of the Jabbok river, both cities being located in northern Trans-Jordan. These cities were requested to supply Gideon's army (here called cam) with "loaves of bread", the intention most likely being food provisions in general, especially as the text emphasizes that the Israelite soldiers were faint with hunger. Gen. 25:29-30 (concerning Esau, coming from the field being 'ayef and he commands Jacob to feed him).
You have one Patriarch commanding another Patriarch, this doesn't go over well. Also, note that Biblical literature only names the most important characters in play (this does happen in other religious texts as well).
Concerning Ishmael you'd have to look at ethnographic names, which Ishmael is included. A genealogy embraces twelve sons, like the Israelite and Ishmaelite systems, consisting of two descent groups—eight sons by his wife Milcah, and four sons by his concubine Reumah. The twofold division here is seemingly geographical: the wife's descendants represent tribal and geographical entities in the Patriarchs' ancestral homeland in Mesopotamia, whereas the sons of the concubine represent place-names in northern Canaan. Nahor's genealogy is quite similar to the Israelite model where Jacob (alias Israel) is the founding ancestor of twelve sons (or tribes) who are born of two wives, Leah and Rachel, or their respective handmaidens, Zilpah and Bilhah
This is where we start dividing tribes up, but this phenomona doesn't just happen with the Israelite's, we see this in other cultures as well, where tribes are made and procolomations are made as to who is worshiping the "true God".
The Akhenaten for example exodus from Egypt as they are Egyptian Monotheists, Amunhotep (IV), Akhenaten ruled Egypt for a mere fourteen years (ca. 1352-1338 BCE), a relatively short reign by the standards of the day during the Amarna Period. We see a separation of Egyptian Monotheists from Polytheistic Egypt. The Akhenaten will claim they worship "the one true God" who is thought to be Aten only in Egyptian Akhenaten religious writ.
Hence, tribes are created and separations happen to claim a name for the "one true God", but as I wrote earlier the term "God" is a misnomer, it is a broad generalization. Whether it is Christian mythology, Sumerology, Assyriology, etc... the term or noun "God" is only widely applied.