Solution to Dirac equation

It is a question of degrees of freedom. A solution to Dirac equation with 1$times$4 components has precisely enough freedom for describing a single (without any "internal" quantum numbers) Dirac fermion which has only two states with different spin orientations. However, if your fermion field has additional quantum numbers (like color charge in strong interactions), the wavefunction really could be thought of as a matrix as you suggested (and if there are several quantum numbers, it will become a multidimensional array). But it's completely the same description as just writing a set of equations for each of 1$times$4 wavefunctions corresponding to fermions with particular quantum numbers.