The City of Bells is a true city-state. They have no real authority beyond the city itself, and they trade for everything they need. There aren't Kharbranthian farmers, for example. If commerce were to fail, the city would flat-out collapse.
They do have their own language, as hinted at in this chapter, but it's very similar to Alethi and Veden. I consider the three languages to really be dialects of Alethi, and learning one is more about learning new pronunciations as it is about learning new words. (Though there are some differences in vocabulary.) I would put them even slightly closer than Spanish and Portuguese in our world.
The city origins are a little less proud than they'd tell you. Kharbranth was a pirate town, a harbor for the less savory during the early days of navigation on Roshar. As the decades passed, however, it grew into a true city. To this day, however, its leaders acknowledge that they're not a world power—and might never be. They use games of politics, trade, and information to play Jah Keved, Alethkar, and Thaylenah against one another.