You have chosen an example where the moduli space (a projective space) is a homogeneous space. So, geometrically, all the objects it parametrises are "the same", and the nature of the moduli space merely confirms that.
Perhaps it should be said first, therefore, that moduli spaces are not always homogeneous spaces. Not all points on the moduli space look the same, and therefore questions arise. This is seen classically for elliptic curves, where the typical automorphism group (preserving the identity) of an elliptic curve is of order 2, but in a few cases it may be of order 4 or order 6. Does this show up in the moduli space? Yes, when you construct it in the classical way from a fundamental domain in the upper half-plane. Moral: if there are "special" points in the moduli space, there is a geometrical reason they are special.
There are actually three levels to look at: the structure of the moduli space qua space (manifold-like, let's say, for complex geometry); for sophisticates using scheme theory the so-called infinitesimal structure in the sheaf given on the space; and the "moduli" themselves, such as the classical j-invariant, namely the parameters used to describe the space. It depends from what direction you are coming, but certainly for arithmetic special values of the moduli read back in an interesting way.