Growing up in Jersey we called them jitneys. And no, they don’t have them in other cities.
Yes, the jitneys. There is the Hampton Jitneys which transport folks from NYC to the Hamptons during summer months. Those jersey buses are better than NJ Transit.
Jitneys could be found all over the country but they've largely fallen out of favor.
There is in Atlanta. An informal system of cash only mini buses without fixed routes and a focus on Spanish language.