Excitations & Pentagon axiom in algebraic theory for anyons

I have been reading the anyon theory by Kitaev and Wang. I have two possibly related questions:

1. Why is the Pentagon equation/axiom sufficient for characterizing associative relations?
2. Are there anyon theories with more than two elementary (non-composite) excitations?

Thanks for the excellent comments by user @AccidentalFourierTransform! My first question may be formulated in this way:

1. Are there physical reasons that enforce the Pentagon axiom? Why do we not consider axioms with more than four labels/excitations? (for example, in the string-net theory, the associative relation of interest is constructed to be a "tensor product" of four "spaces"; what could be the anyon analog of this special requirement?)

My impression has been that there is no anyon theory with more than two non-composite elementary excitations. This seems to be the reason that the Pentagon equation (together with braiding structures, of course) is sufficient in describing anyons. Is this naive intuition correct? Could someone enlighten me on this subject?

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cross-posted on physics overflow (no answers so far).