From my perspective, book smarts and street smarts aren't a personality characteristic. They're simply words for formal knowledge that you learn in school and informal knowledge that you pick up elsewhere. To the degree that they're an indicator of class, it's usually just a matter of someone either fetishizing or looking down on the other form of learning that they haven't personally experienced.
I've spent a lot of time in factories in Asia, and my favorite person to work with over the years was a really smart guy educated in the US (PhD in manufacturing and/or chemistry, IIRC) who opened a small injection molding factory back home in Taipei. Beyond just being a nice guy, I could always depend on him to be able to solve manufacturing challenges that others would tell me are impossible.
One night after dinner when we were walking through a few local shops, he pointed to some expensive tortoise shell glasses
on the shelf and asked "do you know how those are made?" I had no idea. He went on to explain that in school they teach you that you need two colors of plastic, some fancy expensive hot-runner machinery, and a mixer that coarsely combines the molten plastic right at the final moment before it squirts into the tool. You have to get the injection temperatures, pressures, and timing just right or it doesn't turn out, so there's often lots of waste.
Then he grinned and explained that nobody does it that way in China. In most small factories, they just shoot the glasses in yellow and pay a guy with an eye dropper to sprinkle soy sauce on the hot plastic as soon as it comes out of the mold to dye it brown in spots. Most people can't tell the difference. I gathered that he was happy to offer either method depending on my budget and just how particular I was about things being done the American way or the Chinese way.
Book smarts vs. street smarts. I didn't earn a degree in manufacturing that day, and I also didn't suddenly become a local factory hustler. But after that conversation I had both types of smarts on that particular topic.