Does that mean zinc is important for muscle building and copper for organ function?
Not necessarily. Proteins (specifically certain amino acids) are most important for muscle building. Copper has a number of functions, partly some would support organ health, but it isn’t “this is for that, and that’s for this” type scenario.
I don’t know about animals, but for humans most dietary copper is filtered, stored, and excreted by the liver. For example, there’s a rare condition called Wilson’s disease in which the gene responsible for copper transport is defective and causes toxic accumulation of copper in the liver, kidneys, nervous system, heart, and corneas.