  The exact critical Casimir amplitude is derived for anisotropic systems
within the $d=2$ Ising universality class by combining conformal field theory
(CFT) with anisotropic $\varphi^4$ theory. Explicit results are presented for
the general anisotropic scalar $\varphi^4$ model and for the fully anisotropic
triangular-lattice Ising model in finite rectangular and infinite strip
geometries with periodic boundary conditions (PBC). These results demonstrate
the validity of multiparameter universality for confined anisotropic systems
and the nonuniversality of the critical Casimir amplitude. We find an
unexpected complex form of self-similarity of the anisotropy effects near the
instability where weak anisotropy breaks down. This can be traced back to the
property of modular invariance of isotropic CFT for $d=2$. More generally, for
$d>2$ we predict the existence of self-similar structures of the finite-size
scaling functions of $O(n)$-symmetric systems with planar anisotropies and PBC
both in the critical region for $ n \geq 1$ as well as in the
Goldstone-dominated low-temperature region for $ n \geq 2$.
