BB was a name for a specific shogun ammo pellet/bead size (lead or zinc/copper-coated steel).
Shotgun ammo pellet/bead sizes

"Originally, the first BB guns had a bore size of approximately 0.180 inches. This is close to the shotgun pellet size BB, which falls in between shot size B and size BBB. Hence, little boys used lead BB shot in their guns. Later, Daisy standardized lead shot to 0.175 inches and called it air rifle shot."