I felt that it's important to note that not all universality classes are determined solely by spacial dimension and symmetry, even when the interactions are local.
The easiest model that exemplifies this is probably the Ashkin-Teller model in 2D. It has a continuously varying (thus infinite variety of) critical exponents (universality classes) depending on the value of the parameter in the Hamiltonian. The symmetry does not change for different values of the parameter, so this is a counter example to the often made claim that universality classes are determined only by dimensionality and symmetry.
The critical exponent values written in the wikipedia chart with "Ashkin-Teller" only corresponds to ONE point of this continuum of criticalities.
Of course, this kind of examples are rare, and usually universality classes could indeed be predicted from the spacial dimension and the symmetry. But I think it's important to realize that the claim is not a rigorously proven theorem, but is more like a guideline.
There are even recent works on trying to find a non-Ising $Z_2$ symmetry breaking universality class (https://arxiv.org/abs/1803.00578).
Also, to add to the list, I think the cubic-anisotropic Heisenberg universality class is one that is easy to be mistaken with the normal isotropic Heisenberg universality class. The critical exponents are close anyway, and it's actually hard to numerically distinguish them, but RG calculations say that the anisotropy perturbation is relevant, meaning that it flows to a separate fixed point.