No. The main difference between grouping is that phytoplankton receives most of energy by environment(usually Sun) without consuming other plankton and zooplankton does exactly that - it consumes other plankton(both phytoplankton and zooplankton).

Phytoplankton does not belong to Plant kingdom and neither do Zooplankton to Animal kingdom. Some Zooplankton are animals, which is subbranch of Eukariota to which most zooplankton belongs. Some of phytoplankton, like cyanobacteria are not considered to belong to plant kingdom, also most of plankton are supergrouped together with plants where most of them are not plants, but also belong to Eukariota. TL;TR: Animals and Plants belong to Eukariota, to which belong all plankton - even if most of them does not belong neither to Plants nor Animals.