Our little league was the Pony league set up. We had sponsors like Moose Lodge, funeral homes, banks, local pizza joints, athletic apparel stores, etc. My hometown even hosted the Colt League World Series (15-16 years old) for several years, which actually drew great crowds.
My kids played a few years of little league in our current city in Northern Virginia, and they typically use the major league names. Although one season, the Nationals sponsored the league and every team was the Nationals, just different colors.