There's no significance really. It's just a stereotypical lunch bag. I think most people use coolers these days, but back in the day you'd pack your sandwhich, chips, and drink in a brown paper bag to take to work.
It's just a generic term for a (usually low-cost) packed lunch from home, as opposed to going out and eating for lunch. It comes from how cheap lunches were packed in a brown paper back in the old days.
Brown bags are the classic container used to pack a lunch for a child attending school in the US. That practice dates back many generations, although I'm sure it's becoming rare in the last few decades.
in the context of a lunch container it conveys "nothing special", inexpensive, utilitarian, practical as opposed to fancy , expensive, extravagant "three martini" lunch
What's the significance of a lunch bag being brown in the US?