This is an assumption, but in the case of the of the Ising model it is an obviously correct one. The statement that $m$ is the same at all sites is the assumption that the state of the system respects translational invariance. In general a system may spontaneously break translational symmetry, just as it breaks the $s\rightarrow-s$ symmetry. So when you do the usual mean field theory you are assuming that the system maintains the symmetry.
Take for example the Ising model on a square lattice but with an antiferromagnetic interaction, so that neighboring spins want to anti-align. In this case, at zero temperature, the ground state has $s=+1$ on half the sites and $s=-1$ on the other half. This spontaneously breaks the translational symmetry.
Let's say you didn't notice this and proceeded with the derivation as usual. So you assume the magnetization is equal on every site and went through the mean field theory. Nothing would break, mathematically speaking. However, you would find that there is no spontaneous symmetry breaking and that $m=0$. This is wrong. You get the wrong answer because you tried to approximate the system with a ferromagnetic state, when you should have used an anti-ferromagnetic.
Moral: All other problems aside, mean field theory is only as good as the state you are using. Even worse, if you use the wrong state, the mean field theory will give you no idea that you made the wrong choice.