  We study spatial anisotropy effects on the bulk and finite-size critical
behavior of the O$(n)$ symmetric anisotropic $\phi^4$ lattice model with
periodic boundary conditions in a $d$-dimensional hypercubic geometry above, at
and below $T_c$. The absence of two-scale factor universality is discussed for
the bulk order-parameter correlation function, the bulk scattering intensity,
and for several universal bulk amplitude relations. For the confined system,
renormalization-group theory within the minimal subtraction scheme at fixed
dimension $d$ for $2<d<4$ is employed. For the case of cubic symmetry and for
$n=1$ our perturbation approach yields excellent agreement with the Monte Carlo
(MC) data for the finite-size amplitude of the free energy of the
three-dimensional Ising model at $T_c$ by Mon [Phys. Rev. Lett. {\bf 54}, 2671
(1985)]. Below $T_c$ a minimum of the scaling function of the excess free
energy is found. We predict a measurable dependence of this minimum on the
anisotropy parameters. The relative anisotropy effect on the free energy is
predicted to be significantly larger than that on the Binder cumulant. Our
theory agrees quantitatively with the non-monotonic dependence of the Binder
cumulant on the ferromagnetic next-nearest neighbor (NNN) coupling of the
two-dimensional Ising model found by MC simulations of Selke and Shchur [J.
Phys. {\bf A 38}, L739 (2005)]. Our theory also predicts a non-monotonic
dependence for small values of the {\it antiferromagnetic} NNN coupling and the
existence of a Lifschitz point at a larger value of this coupling. The
nonuniversal anisotropy effects in the finite-size scaling regime are predicted
to satisfy a kind of restricted universality. The tails of the large-$L$
behavior at $T \neq T_c$ violate both finite-size scaling and universality.
