The previous understanding of the quantum spin liquid as a ground state of spin systems with spin rotation symmetry is not only out-of-date but also misleading. In modern language, quantum spin liquids are classified as symmetry enriched topological(SET) states, which possess anyon excitations carrying fractionalized symmetry charges, meaning that the anyons transform projectively under symmetry actions. The symmetry does not need to include the global spin rotation symmetry SO(3). Therefore, a quantum spin liquid does not need to preserve the spin-SO(3) symmetry in the general sense.
The defining property of the spin liquid is the intrinsic topological order (or the quantum order for gapless spin liquid). The Kitaev spin liquid possesses the $\mathbb{Z}_2$ topological order, which makes it a spin liquid, although the spin rotation symmetry is explicitly broken on the Hamiltonian level. In the current classification scheme, the Kitaev spin liquid is an SET state with $\mathbb{Z}_2$ topological order enriched by space group (translation, $C_6$ rotation and reflection) and time-reversal symmetry, and therefore satisfies the modern SET definition of the quantum spin liquid.
Of cause, you may restrict the discussion of spin liquid to the spin rotational symmetric cases, i.e. the spin-SO(3) symmetric spin liquid, which is just a subclass of all spin liquids, and indeed Kitaev spin liquid does not belong to this subclass. However, it is possible to write down a variation of the Kitaev model which is spin-SO(3) symmetric, and the resulting ground state is a spin-SO(3) symmetric spin liquid.