The binding energy of a bipolaron in an ionic crystal increases substantially in the case of strong anisotropy of the effective masses of the free carriers of the easy plane type or easy axis type. In the second case the polaron is cigar-like in shape and the coaxial configuration of bipolarons is energetically favorable. In this case a significant gain in the binding energy and in the width of the region of existence of the bipolaron, with respect to the dielectric constant and the magnitude of the electron-phonon interaction constant, compared with an isotropic crystal, is obtained only for quasi-two-dimensional, or layered, and quasi-one-dimensional, or chainlike, crystals. This work shows that a significant gain in the binding energy can be obtained by taking into account the anisotropy of the dielectric constant of the crystal and localization of the electron wave functions in directions perpendicular to the layers and chains of atoms.