When Americans use the term cat in the most generic sense, they are referring to the American Shorthair, and the vast majority of American cats belong to this breed. Alley cats are American Shorthairs, and so is the lion’s share (pun intended) of household pets, whose owners usually don’t particularly care about things like pedigrees.
However, there are pedigreed American Shorthairs, and “the papers” are required to enter them into cat shows. But many cat shows now have a household pet category that allows nonpedigreed cats to compete.