The 's' and 'd' orbitals are related to different eigenvalues of the angular momentum operator (l = 0 for 's' and l = 2 for 'd'). Therefore, their angular parts are described by different spherical harmonics. This harmonic is simmetric for l = 0 and not for l = 2.

In a relativistic case it is more tricky since the angular momentum operator does not commute with the Hamiltonian. In this case the spherical spinors are to be used instead of spherical harmonics.

Think about it this way: You want to represent an arbitrary vector in terms of its components.  You can choose i,j and k, but you can also choose i, (j+k)/sqrt(2), (j-k)/sqrt(2). Eigenstates like s and d orbitals are just that: Base vectors. And they cannot be the same, else you would not have linear independence Which means thay will have different properties, including density distributions.