The fine structures of solar radio bursts reveal complex dynamics in the corona, yet the observed characteristics of these subsecond bursts are additionally complicated by radio wave scattering in the turbulent solar corona. We examine the impact of anisotropic turbulence in radio wave propagation simulations with nonradial magnetic field structures in shaping the morphology, time characteristics, and source positions of fine structures. The apparent sources are found to move along the direction of the magnetic field lines and not along the density gradient, whereas the major axis of the scattered source is perpendicular to the local magnetic field (the scattering anisotropy axis). Using a dipolar magnetic field structure of an active region, we reproduce observed radio fine-structure source motion parallel to the solar limb associated with a coronal loop and provide a natural explanation for puzzling observations of solar radio burst position motions with the Low Frequency Array. Furthermore, the anisotropy aligned with a dipolar magnetic field causes the apparent-source images to bifurcate into two distinct components, with characteristic sizes smaller than in unmagnetized media. The temporal broadening induced by scattering reduces the observed frequency drift rate of fine structures, depending on the contribution of scattering to the time profile. The findings underscore the role of magnetic field geometry and anisotropic scattering for the interpretation of solar radio bursts and highlight that anisotropic scattering produces more than a single source.