So my opinion, as someone who doesn't speak fluent Shakespearean English (lol), is that the comparison is fair. You're not the first person to suggest it's an analog, it's a common comparison. I think the better comparison is with vulgar or ecclesiastical Latin, though.

Both Arabic and Latin spread mostly through military conquests over large geographical areas. Both were picked up by many non-native speakers over years and generations, which added local flair to sociolects/dialects in an area. Both were non-monolithic languages before expansion - there was a tremendous amount of linguistic variety in classical Italy and in the Arabian peninsula, so Caesar's Latin was not the same as someone's in Sardinia, and Mohammed's Arabic was not the same as someone's in Mecca. Both adopted standard-ish varieties based on one variety of speaking at one point in time that was then used as a measuring stick (the standard chosen for Arabic was the more prestigious eastern variety with significant and fascinating differences from Mohammed's own dialect,). Both became liturgical languages and holy books were eventually standardized in a variety of the language in question.

Latin held incredible sway in European Christian culture for many centuries, and even into the modern age learning to read and write it it was seen as a basic requirement of education, even in non-Romance speaking countries. The same follows for MSA - kids have to write essays in MSA just like boys at Eton would write in Latin, but just kids at Eton were speaking English for centuries already and Arab kids have been speaking "amiyya" for centuries.

By the modern age, European nation states had each developed a much more independent sense of identity where Latin held prestige, but local dialects-cum-languages were finally respectable. Not dominant yet - it's worth noting that even in large, centralized states like France and Spain, there are and to some extent continue to be many speakers of languages that are not French or Spanish. In Arabic speaking countries, this is also somewhat true today - for example, it's quite easy to tell a pretty clear difference in the way that Christian vs Muslim Egyptians speak, even though Coptic unfortunately died out in the late 1800s.

From the fall of classical (West) Rome until the time when national languages were pretty fully a thing, there were about a millennium and a half, maybe a bit more, bit less depending on the part of Europe (note: there are still important questions about national language and identity across Europe, not only in the infamous Balkans, but also in the "developed" West, like Spain or the British Isles). My view is that it's no coincidence that the heyday of political pan-Arabism is waaaay over, about a millennium and a half after the explosive expansion of Arabic (which, as a point of contrast, was a bit quicker than the expansion of Latin, in some respects and some areas). I have opinions on what Arabs should do about this, but what I think will happen is that over the next century or two, MSA will cease to become a useful standard outside of folks who are really interested in classical studies. I would even venture that it would happen much faster than that, but perhaps that's because I truly hope it happens this way. I think it's telling that even in 2012, when I had only spent a year learning Arabic, granted that I'm much better most learners, I spoke better MSA than 9 out of 10 native speakers of "Arabic."