The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.