In this paper, we design a novel H-shaped photonic crystal fiber (PCF) surface plasmon resonance (SPR) sensor, which is composed of a dual-sided U-shaped groove and an Ag-MoS2 coating. The proposed sensor exhibits an asymmetrical structure with birefringence, which provides a possible way to detect the refractive index (RI) of analytes with dual-polarization. Based on the finite element method (FEM) simulations, the influences of the structural parameters on the sensing performances are discussed. For y-polarization, the proposed sensor detects the changes of RI from 1.37 to 1.41 with a maximum wavelength sensitivity of 11,700 nm/RIU, while for x-polarization, RI changes from 1.29 to 1.35 with a maximum sensitivity of 7,800 nm/RIU. Thus, a PCF-SPR sensor with dual-polarization and wide RI detection range is achieved, which holds significant promise for applications in biology, medicine, and food safety.