Hi, so the answer is that it's incorrect that dmrg (with open boundary conditions and without conserving Sz-parity symmetry) will give an entropy of ln2, except perhaps on very small system sizes.
You are right, of course, that there is a two-fold ground state degeneracy but this means that dmrg has succeeded if it gives any linear combination of these two ground states, since it's goal is just to minimize the energy. Typical DMRG calculations result in ground-state combinations which have the minimal entanglement entropy. For the ferromagnetic Ising chain this is either all spins up or all spins down, not a linear combination of the two.
The reason I mention small system sizes, is that, for small transverse field there is a tiny reduction in energy due to have a linear combination ("cat state") of both all up and all down ground states, but this energy 'gain' goes away exponentially quickly as a function of system size, so it's typically quite negligible and the truncations that happen inside DMRG will remove this solution.