No. Formal Arabic plays the same role as Formal English. It's the language of business, government, media, education and the learned classes.
Every time you mind your grammar and articulate something for polite company, more than you would have for buddies, you're speaking the English equivalent of "Formal Arabic".
I don't know where people get this meme about Arabic being a dead prestige language.
Once upon a time, in Europe, all the important books (scientific and literary) were written in Latin, all educated and upper-class people knew Latin, learning grammar meant learning Latin grammar, and coversations on serious topics were held in Latin. (And if you really wanted to show off, you learned Latin and Greek.) Latin is a dead prestige language now (except perhaps among Catholic priests), but it wasn’t then.
My point (which may still be incorrect) is that formal Arabic now plays a social role similar to the role that Latin played then. Sorry if I did not communicate that clearly.
And you would absolutely be correct, after clarification. Only difference is that when Latin was supreme, literacy was scarce and only available to the wealthy. Today, schooling is the only thing available to the poor. The average Arab is underemployed and over educated. While among the wealthy, Western languages and just complete illiteracy is popular (see any "heir" give a speech and you will see a man at war with grammar.)
Every time you mind your grammar and articulate something for polite company, more than you would have for buddies, you're speaking the English equivalent of "Formal Arabic".
I don't know where people get this meme about Arabic being a dead prestige language.