The modern Middle-East was largely created by the French and the British, who occupied most of it in the First World War and then drew the maps according to how they understood the local politics. The two communities which had contributed to their war effort were the Jews and the Arabs, and they were represented at the Treaty of Versaille, where the newly formed League of Nations rubber stamped the boundaries the British and French had agreed.
Major groups like the Kurds had no representation, had yet to develop a nationalist movement and were not players.
In the years that followed, a notion developed that the Middle East was an Arab area and anyone who spoke Arabic was deemed to be an Arab. This generally worked well for Christians and Moslems but the status of Jews was unclear for a variety of reasons.
One well worn argument says that Jews aren't Arabs because they lived in the Middle East (outside Arabia) before the Arabs conquered it. But that just raises more problems, because most Christian communities also predate the Arabs but are still regarded as Arab. The Arabs of Arabia also predate the Arab conquest and if that is the case, can Arabic speakers outside Arabia really be regarded as Arabs, or are they just subjugated people who adopted the Arabic language? To put it differently, if Indians and Nigerians speak English, does that make them English?
In practice most of the Arab world's Jews left for Israel (for a variety of reasons) and don't consider themselves Arabs, so the issue is a marginal one. Self-identification is a key factor in national identity. If you think you're an Arab and you speak Arabic then you are. If you think you're an Israeli and don't want to be an Arab then you are not, but self-identification is also flexible and people's minds can change.
Altogether, some 35% of Israeli Jews originate from Arabic speaking countries and therefore, are as entitled to call themselves Arabs, as any Arab-American such as say, Edward Said. Their culture is Jewish but also Arab and common features can be discerned if you choose to make that comparison.
An additional 18% of Israelis are Palestinian-Arabs (and generally self-identify as such) and so if we add the two together than more than 50% of the population of Israel is Arab.
By way of contrast, the other European created regional state, Iraq, is 75% Arab, Algeria is about 75% Arab (although it is debatable how Arab the Berber population is), Egypt is 85-90% Arab (Coptic Christians, like Jews, are not defined as Arabs) and Syria is 90% Arab.
Finally there is an inter-marriage issue: about 1 in 4 Israeli marriages is between Jews of European origin and Jews of Middle-Eastern origins, and given time all Israelis will be descended from both ethnic groups.