The study of measurements in quantum mechanics exposes many of the ways in which the quantum world is different. For example, one of the hallmarks of quantum mechanics is that observables may be incompatible, implying among other things that it is not always possible to find joint probability distributions which fully capture the joint statistics of multiple measurements. Instead, one must employ more general tools such as the Kirkwood-Dirac quasiprobability (KDQ) distribution, which may exhibit negative or nonreal values heralding nonclassicality. In this paper, we consider the KDQ distributions describing arbitrary collections of measurements on disjoint components of some generic multipartite system. We show that the system dynamics ensures that these distributions are classical if and only if the Hamiltonian supports quantum Darwinism. Thus, we demonstrate a fundamental relationship between these two notions of classicality and their emergence in the quantum world.