I would interpret things like "has richer structure than just X" to mean that there's more there that you get for free, in some sense. For instance, in your example, the set of solutions to a system of polynomial equations at first seems to just have a cardinality, which is the only thing you can say automatically about a set. However, in this case, it's not just any set, by virtue of being solutions to some polynomial equations over a ring, it has a structure of a scheme (variety if the ring is a field and the equations are nice) where, without making any choices at all, we can make it into something more than just a set. In some cases (elliptic curves) the structure is yet richer, and is in fact also a group in a natural way.
At the least, things like this are how I generally hear the phrase used. But generally it's used informally to say that something is more interesting than a random object that looks like it at first glance, or that an object is very interesting (richness of structure of the absolute Galois group) and in nontrivial ways.