The answer is that it did.

The Arabic that is the official language in all Arab countries today is called "Modern Standard Arabic," commonly called "MSA." This is Classical Arabic taken straight from the Quran updated for modern times. It is the only written form of Arabic, and is the form of Arabic spoken on the news, in academic circles, and in politics.

In this sense, MSA indeed performs a similar function today that Latin performed in Europe in the Middle Ages, and much like Latin in the Middle Ages, MSA today is only spoken by a small sliver of political and academic elite. All Arabs understand it, but they don't speak it. In fact, there are no native speakers of MSA. Instead, much like the Chinese, all Arabs speak a wide variety of "dialects," many of which are only partially mutually intelligible with each other, and some not at all. However at least in the case of China, Standard Chinese (Mandarin) is actually a native spoken language of many, and is acceptable (if not expected) to be used in all spheres of life, at least alongside dialect. MSA meanwhile in the Arab world has almost no penetration into daily life at all. Though they are grouped together for political and cultural reasons, the so-called "dialects" of Arabic are only slightly more similar to each other than Dutch-German, or Italian-Spanish.

Many language learning courses advertise MSA as the "lingua franca" of the Arab world spoken by "300 million people," or as the fifth most spoken language in the world and so on. This is not true. If two Arabs meet who speak different dialects, almost always they will try to simply muddle through with their own dialects. Their success will depend greatly on the distance between dialects, but it is possible - especially when speakers deliberately moderate their speech. Many Arabs have at least some experience hearing other dialects and accents, which eases communication somewhat (much like a Spaniard who consumed Italian media could understand Italian). However, sometimes the dialects are simply too far apart for mutual comprehension, such as the Maghrebi dialects of Tunisia, Algeria and Morocco, which even most eastern Arabs will admit are incomprehensible to them. Yet even then Arabs will often switch, not to MSA, but to a western language like French or English to communicate with each other.

Modern Standard Arabic is taught because it is the only written form of Arabic, which follows because it is the language in which the Quran is written and is therefore the tongue of the Prophet Muhammad. This makes Classical Arabic not just a politically unifying language but literally holy. Dialects have no official written forms and remain uncodified, due to Arabs refusing to accept them as "languages." In this sense, Arabs may be said to read in one language and speak in another - a situation called "diglossia."

Note that this is not at all the answer you will receive from many native Arabs. Much the same way that, due to political animosity, Pakistanis and Indians who speak Urdu and Hindi will insist they cannot understand each other despite almost all linguists concluding colloquial Urdu and Hindi are the same language, Arabs will insist they can all understand each other even when they cannot. This is because the issue is bound up tightly with ideas of Arab political and cultural unity, along with the widespread view promoted in Arab society that only Classical Arabic (and by extension MSA) is "Real Arabic," and that dialects are all just slang.

All this means that the "Official Language" of all Arab countries is a language that almost nobody actually speaks or uses in daily life. An example of this nationalistic protectiveness of MSA is when the Masri (Egyptian dialect) Wikipedia project was first begun, it was furiously condemned by many Arabs who declared that Masri was "not a language," and many accused the project of being part of a Coptic-Christian plot to undermine Arab unity.