We numerically study the anisotropic Turing patterns (TPs) of an activator-inhibitor system described by the reaction-diffusion (RD) equation of Turing, focusing on anisotropic diffusion using the Finsler geometry (FG) modeling technique. In FG modeling, the diffusion coefficients are dynamically generated to be direction dependent owing to an internal degree of freedom (IDOF) and its interaction with the activator and inhibitor. Because of this dynamical diffusion coefficient, FG modeling of the RD equation sharply contrasts with the standard numerical technique in which direction-dependent coefficients are manually assumed. To find the solution of the RD equations in FG modeling, we use a hybrid numerical technique combining the Metropolis Monte Carlo method for IDOF updates and discrete RD equations for steady-state configurations of the activator-inhibitor variables. We find that the newly introduced IDOF and its interaction are a possible origin of spontaneously emergent anisotropic patterns of living organisms, such as zebra and fishes. Moreover, the IDOF makes TPs controllable by external conditions if the IDOF is identified with the direction of cell diffusion accompanied by thermal fluctuations.