May I ask why? I as an atheist did memorize some part of the Bible for the fun of it but I understand the language.

Because it is important to me as a Muslim. Also it is challenging, interesting, to try and memorize an entire book.

... but wouldn't it make to learn classical Arabic alongside with that?

In Islam there's a certain reverence for memorizing the Quran, unlike in Christianity.

In fact, being a Hafiz or your child being a Hafiz is a point of pride.

This in part goes back to Islamic lore/history.

Another part is that there is the belief that there are rewards associated with, being accompanied by angels iirc.