The investigation of the converging shock-induced Richtmyer-Meshkov instability, which arises from the interaction of converging shocks with the interface between materials of differing densities in cylindrical capsules, is of significant importance in the field of inertial confinement fusion (ICF). The use of converging shocks, which exhibit higher efficiency than planar shocks in the development of the RMI due to the Bell-Plesset effects, is particularly relevant to energy production in the ICF. Moreover, external magnetic fields are often utilized to mitigate the development of the RMI. This paper presents a systematic investigation of the anisotropic nature of the Richtmyer-Meshkov instability in magnetohydrodynamic induced by the interaction between converging shocks and perturbed semicylindrical density interfaces (DI) based on numerical simulations using Athena++. The results reveal that magnetic fields with β=1000, 100, and 10 (β is defined as the ratio of the plasma pressure to the magnetic pressure) lead to an anisotropic intensification of magnetic fields, anisotropic accelerations of various shock waves [including the converging incident shock (CIS), transmitted shock (TS), and reflected shock (RS)], and anisotropic growth of the DI with subsequent anisotropic vorticity distribution. Upon closer inspection, it becomes evident that these phenomena are strongly interconnected. In particular, the region where the wave front of the CIS impacts the middle point of semicylindrical DI, where the magnetic field is more perpendicular to the fluid motion, experiences a more significant amplification of the magnetic fields. This generates higher-pressure jumps, which in turn accelerates the shock wave near this region. Furthermore, the anisotropic amplification of the magnetic fields reduces the movement of the RMI near the middle point of semicylindrical DI and leads to the anisotropic formation of RMI-induced bubbles and spikes, as well as vortices. By examining vorticity distributions, the results underscore the crucial role of magnetic tension forces in inhibiting fluid rotation.