  Universality is a common quality in different physical parameters that is
rooted in the deep nature of physical systems. Scaling relation is a typical
universality for critical phenomena around a quantum phase transition, while
topological classification provides another type of universality essentially
different from the critical universality. Both classes of universalities can be
present in a single-qubit system with light-matter interactions, as exhibiting
generally in the fundamental quantum Rabi model with anisotropy not only for
linear coupling but also for nonlinear Stark coupling (NSC). In low frequencies
different levels of scaling relations are demonstrated, holding for anisotropic
or/and NSCs, locally or globally. At finite frequencies such a critical
universality breaks down and diversity is dominant. However, common topological
feature of the ground state can be extracted from the node number, which yields
a topological class of universality amidst the critical diversity. Both
conventional and unconventional topological transitions emerge, with their
meeting, which never occurs in linear interaction, enabled by the nonlinear
coupling to form topological quadruple points which are found to be
spin-invariant points. Sensitivity analysis indicates that the NSC can be
another approach to manipulate topological transitions in addition to coupling
anisotropy.
