Topological interface states (TISs), known for their distinctive capabilities in manipulating electromagnetic waves, have attracted significant interest. However, in conventional all-dielectric one-dimensional photonic crystal (1DPC) heterostructures, TISs strongly depend on incident angle, which limits their practical applications. Here, we realize an angle-independent TIS in 1DPC heterostructures containing hyperbolic metamaterials (HMMs) for transverse magnetic polarized waves. We begin with the design of two kinds of angle-independent photonic bandgaps (PBGs) in two 1DPCs with symmetric unit cells based on the phase-variation compensation effect. From the Zak phases of the upper and lower bands, the topological properties of PBGs in two 1DPCs are different. By harnessing different topological properties, we can realize an angle-independent TIS in the heterostructure composed of these two 1DPCs. Moreover, we further discover that the angle-independent property of the TIS is robust against the layer thickness variation due to topological protection, making the experimental realization of the angle-independent TISs more feasible. It is noted that the TISs still depend on the incident angle under transverse electric polarized waves since the iso-frequency curve of the HMM is a circle. Empowered by the polarization-dependent property of the TIS, we design a wide-angle polarization selector with an operating angle range up to 45.9°. Our work provides a viable route to realizing angle-independent TISs with substantial angular tolerances under current experimental conditions, which facilitates the design of optical devices including polarizers, filters, and sensors with robustness against disorder.