I'm following this solution of
$$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$
And the question is whether its solution
$$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t e^{-\kappa(t-u)}\,dW_u$$
unique? Of course, (1) is a version of a more general SDE, for which uniqueness is established. However, in these particular setting construction of the solution should imply uniqueness. Also for any other solution $X'$ and $D=X-X'$, (1) implies that
$$D_t=-\kappa\int_0^t D_s \, ds, \quad D_0=0,$$
which is just an ordinary DE so that $D(t)=0$ is the unique solution...