The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution.

The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution.

The Sato-Tate distribution and the Wigner semicircle distribution are the same.

Combinatorially, this comes from the fact that the moments of both those two distributions are the Catalan numbers.

We can categorify this as follows: If $A$ is a self-dual element of a monoidal abelian category over a field, then $Hom(1, A^{\otimes n})$ is a vector space. (By self-duality, this is also equal to $Hom(A^{\otimes a}, A^{\otimes b} )$ for any $a,b$ such that $a+b=n$.)

This categorifies the $n$th moment in the following way: If $G$ is a compact Lie group, and $A$ is a self-dual representation in the category of representations of $A$, then the dimension of $Hom(1, A^{\otimes n})$ is the $n$th moment of the trace of a random element of $G$ acting on $A$. So in the case $G=SU_2$, $A$ the standard representation, the dimension is the Catalan numbers.

On the other hand, if you take a free monoidal abelian category on one self-dual generator, the moments are the Catalan numbers. (One has to take dimensions, not just over the base field, but over the ring generated over the base field by the element of $Hom(1,1)$ defined by the canonical morphism $1 \to A \otimes A \to 1$.)

Since the category of representations of $SU_2$ is certainly a monoidal abelian category, we have a functor from the second category to the first that sends the free generator to the standard representation. So we have a morphism from the free moment vector spaces to the moment vector spaces of $SU_2$.

I believe I can show by an explicit combinatorial argument that this is an isomorphism, categorifying this identity between the two definitions of Catalan numbers.

Hence the free monoidal abelian category on one self-dual generator is equivalent to the category of finite-dimensional representations of $SU_2$.

Is there some conceptual reason why this is so?

Note that the limit of the category of representations of $USP_{2n}$ as $n$ goes to $\infty$ is the free symmetric monoidal abelian category on one self-dual generator.