  Analytic predictions have been derived recently by V. Dohm and S. Wessel,
Phys. Rev. Lett. {\bf 126}, 060601 (2021) from anisotropic $\varphi^4$ theory
and conformal field theory for the amplitude ${\cal F}_c$ of the critical free
energy of finite anisotropic systems in the two-dimensional Ising universality
class. These predictions employ the hypothesis of multiparameter universality.
We test these predictions by means of high-precision Monte Carlo (MC)
simulations for ${\cal F}_c$ of the Ising model on a square lattice with
isotropic ferromagnetic couplings between nearest neighbors and with an
anisotropic coupling between next-nearest neighbors along one diagonal. We find
remarkable agreement between the MC data and the analytical prediction. This
agreement supports the validity of multiparameter universality and invalidates
two-scale-factor universality as ${\cal F}_c$ is found to exhibit a
nonuniversal dependence on the microscopic couplings of the scalar $\varphi^4$
model and the Ising model. Our results are compared with the exact result for
${\cal F}_c$ in the three-dimensional $\varphi^4$ model with a planar
anisotropy in the spherical limit. The critical Casimir amplitude is briefly
discussed.
