"A unitary 1d QFT consists of a Hilbert space and a Hamiltonian. A group acting on a 1d QFT is a group acting on the Hilbert space, commuting with the Hamiltonian. Note that the *data* of an action only involves the Hilbert space. The Hamiltonian is only there to provide a constraint. Moreover, all 1d QFT have isomorphic Hilbert spaces (except in special cases, e.g. in the case of a 1d TQFT, when the Hilbert space is finite dimensional).
A unitary 2d QFT consists of the 0-dimensional and 1-dimensional part of the QFT, along with the data of the Stress-energy tensor. An action of a fusion category on a 2d QFT is again something where the *data* only involves the 0-dimensional and 1-dimensional part of the QFT, while the Stress-energy tensor is only there to provide a constraint. The upshot is that it makes sense to act on the 0-dimensional and 1-dimensional part of the QFT. Moreover, I conjecture that all 2d QFTs have isomorphic 0-dimensional + 1-dimensional parts (except in special cases, e.g. in the case of a chiral CFT)."