We study identifiability of finite mixtures of Dirichlet distributions on the interior of the simplex. We first prove a shift identity showing that every Dirichlet density can be written as a mixture of $J$ shifted Dirichlet densities, where $J-1$ is the dimension of the simplex support, which yields non-identifiability on the full parameter space. We then show that identifiability is recovered on a fixed-total parameter slice and on restricted box-type regions. On the full parameter space, we prove that any nontrivial linear relation among Dirichlet kernels must involve at least $J$ coefficients sharing a common sign, and deduce that mixtures with fewer than $J$ atoms are identifiable. We further report direct non-identifiability implications for unrestricted finite mixtures of generalized Dirichlet, Dirichlet-multinomial, fixed-topic-matrix latent Dirichlet allocation, Beta-Liouville, and inverted Beta-Liouville models.