No, types aren't kingdoms... Did you fail 5th-grade biology or something? Even within each type there are different kingdoms - the Grass type is the best example, as it holds both plants and reptiles (Tropius, Treecko, Chikorita, etc). Likewise with other types, there can be things like foxes in different types (Eevee is normal, its evolutions have different types, Vulpix, etc). There are no types that are one single Class; Normal-types are mostly mammals, but they also have things like Porygon, Ditto, Kecleon, etc. so that doesn't work either.
Even if you try to break Pokemon up into something as small as classes it doesn't fit. Some Pokemon are based off of inanimate objects (Wobbuffet, Shuppet, Unown), ghosts, rocks, and other things that were never alive in the first place and thus do not fit into a scientific classification model. Others are combination of animals that aren't even in the same class (Gligar is a "scorpion bat"; scorpions are arachnids, bats are mammals). And obviously the breeding groups don't help, because... HSOWA. Egh.
Fact is: biology + magical fictional creatures = no.