Has the notion of a unique factorization category been defined and studied?

Question

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a prime object of $\mathcal{C}$ is an object $x$ such that if $x\sim y\otimes z$ (where $\sim$ means is isomorphic to), then $I\sim y$ or $I\sim z$. We call a category $\mathcal{C}$ a unique factorization category if all (non-unit) objects may be written as a tensor product of prime objects uniquely up to reordering the factors. My question is: has this notion been defined in the literature? If so where?