We characterize the biorthogonal ensembles that are both a multiple orthogonal polynomial ensemble and a polynomial ensemble of derivative type (also called a Pólya ensemble). We focus on the notions of multiplicative and additive derivative type that typically appear in connection with products and sums of random matrices respectively. Essential in the characterization is the use of the Mellin and Laplace transform: we show that the derivative type structure, which is a priori analytic in nature, becomes algebraic after applying the appropriate transform. Afterwards, we use the characterization to show that the eigenvalue densities of products of JUE and LUE matrices are essentially the only multiple orthogonal polynomial ensembles of multiplicative derivative type. We also show that the eigenvalue densities of sums of dilated LUE and GUE matrices are examples of multiple orthogonal polynomial ensemble of additive derivative type, but provide other examples as well. Finally, we explain how these notions of derivative type can be used to provide a partial solution to an open problem related to orthogonality of the finite finite free multiplicative and additive convolution of polynomials from finite free probability. In particular, we obtain families of multiple orthogonal polynomials that (de)compose naturally using these convolutions.