  Recently a unified hypothesis of multiparameter universality for the critical
behavior of bulk and confined anisotropic systems has been formulated [V. Dohm,
Phys. Rev. E {\bf 97}, 062128 (2018)]. We prove the validity of this hypothesis
on the basis of the principle of two-scale-factor universality for isotropic
systems. We introduce an angular-dependent correlation vector and a generalized
shear transformation that transforms weakly anisotropic systems to isotropic
systems. As examples we consider the $O(n)$-symmetric $\varphi^4$, Gaussian,
and $n$-vector model. We determine the structure of the bulk order-parameter
correlation function, of the singular bulk part of the critical free energy,
and of critical bulk amplitude relations of anisotropic systems. It is shown
that weakly anisotropic systems exhibit a high degree of intrinsic diversity
due to $d(d+1)/2-1$ independent parameters. Exact results are derived for the
$d=2$ Ising universality class and for the spherical and Gaussian universality
classes. For the $d=3$ Ising universality class we identify the universal
scaling function of the isotropic bulk correlation function from the
nonuniversal result of the functional renormalization group. A proof is
presented for the validity of multiparameter universality of the exact critical
Casimir amplitude in a rectangular geometry of weakly anisotropic systems with
periodic boundary conditions in the Ising universality class. This confirms the
validity of recent predictions of self-similar structures of finite-size
effects at $T=T_c$ derived from conformal field theory. This also substantiates
the previous notion of an effective shear transformation for anisotropic
two-dimensional Ising models. Our theory paves the way for a quantitative
theory of nonuniversal critical Casimir forces in anisotropic superconductors.
