In this study, the scale-dependent shear modulus of pure copper was investigated using a self-developed microscale torsion testing machine and electron backscatter diffraction (EBSD). The shear moduli of copper bars (4 mm diameter) and copper wires (200 μm diameter) were measured and compared. The copper bar exhibited a shear modulus of 45.5 GPa, which aligns well with the Voigt–Reuss–Hill model predictions (46.8–47.3 GPa), attributed to its random texture and isotropic behavior. In contrast, the copper wire showed a notably lower shear modulus of 36.9 GPa. Despite both samples having comparable grain sizes, elastic stiffness, misorientation angles, Schmid factors, and Taylor factors, these microstructural features did not account for the observed modulus difference. EBSD analysis revealed that the copper wire developed strong Normal Direction (ND)//[001] and ND//[111] fiber textures, leading to anisotropic mechanical behavior. Theoretical estimates based on texture components predicted a shear modulus range of 29.7–40.1 GPa for the wire, consistent with the experimental result. These findings indicate that crystallographic texture, rather than grain size, plays a dominant role in determining the shear modulus at the microscale, offering valuable guidance for the development of high-performance flexible electronic devices.