By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "equivalent"; for me (braided) fusion categories are just F-tensors (and R-tensors), in which case it's really "equal" (up to a gauge on the fusion vector spaces)]
I'm asking because there is a very direct physical translation of that statement:
"Two gapped 2+1-dimensional non-chiral topologically ordered systems are in the same phase if their anyon content is the same."
By "in the same phase" here I mean equivalence under a short-range local unitary circuit, or in other words an invertible domain wall. The statement seems so accepted in the physics community that hardly anyone talks about it. Still I never heard a simple argument why it should be so.
For the Turaev-Viro-Levin-Wen fixedpoint models, the physical systems are constructed from fusion categories, and invertible domain walls can be constructed from invertible bi-modules.