For dual typed Pokemon, the primary typing decides how much power you gain your kingdom. After a Pokemon is hatched, if it is Dual typed, it must decide upon what kingdom it will join. If it evolves from a single type to a dual type, it must also decide.
Take Slowking for example: it is primary typed as Water and secondary typed Psychic. If Slowking joined the Water Kingdom, it would have strong influence and power. If it chooses Psychic, it will have next to no power or influence, but will be respected for choosing the "superior" typing.