We investigate a possibility of Hund's metal behavior in the Hubbard model with asymmetric density of states having peak(s). Specifically, we consider the degenerate two-band model and compare its results to the five-band model with realistic density of states of iron and nickel, showing that the obtained results are more general, provided that the hybridization between states of different symmetry is sufficiently small. We find that quasiparticle damping and the formation of local magnetic moments due to Hund's exchange interaction are enhanced by both the density of states asymmetry, which yields stronger correlated electron or hole excitations, and the larger density of states at the Fermi level, increasing the number of virtual electron-hole excitations. For realistic densities of states, these two factors are often interrelated because the Fermi level is attracted towards peaks of the density of states. We discuss the implication of the obtained results to various substances and compounds, such as transition metals, iron pnictides, and cuprates.