Vertically stacked van der Waals (vdW) heterostructures have introduced a unique way to engineer optical and electronic responses in multifunctional photonic and quantum devices. However, the technical challenges associated with the artificially fabricated vertical heterostructures have emerged as a bottleneck to restrict their proficient utilization, which emphasizes the necessity of exploring naturally occurring vdW heterostructures. As one type of naturally occurring vdW heterostructures, franckeite has recently attracted significant interest in optoelectronic applications, but the understanding of light-matter interactions in such layered mineral is still very limited especially in the nonlinear optical regime. Herein, the anisotropic Raman scattering and third-harmonic generation (THG) from mechanically exfoliated franckeite thin flakes are investigated. The observed highly anisotropic Raman modes and THG emission patterns originate from the low-symmetry crystal structure of franckeite induced by the lattice incommensurability between two constituent stacked layers. The thickness-dependent anisotropic THG response is further analyzed to retrieve the third-order nonlinear susceptibility for franckeite crystal. The results discussed herein not only provide new insights in engineering the nonlinear light-matter interactions in natural vdW heterostructures, but also develop a testbed for designing future miniaturized quantum photonics devices and circuits based on such heterostructures.