Откуда взялось слово «коп» и почему, например, в России так не говорят?

Слово "коп" (a cop) - детище английского языка. Это сокращение от существительного "copper", которое произошло от глагола "cop" (to seize, to catch - схватить, налагать арест, ловить).

cop (n.)

"policeman," 1859, abbreviation of earlier copper (n.2), 1846, from cop (v.).

cop (v.)

1704, northern British dialect, "to seize, to catch," perhaps ultimately from Middle French caper "seize, to take," from Latin capere "to take" (from PIE root *kap- "to grasp"); or from Dutch kapen "to take," from Old Frisian capia "to buy," which is related to Old English ceapian (see cheap).