We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.