Why is there no symplectic version of spectral geometry?

The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of projectivized linear symplectic transformations. This group acts transitively on the real points of projective space, so preserves no hypersurface.

From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. From this point of view, that there is no spectral theory of symplectic manifolds is no more surprising than that there is no spectral theory of smooth manifolds. Generally speaking, the symmetries of a second order differential operator, such as a Laplacian, are finite-dimensional, and its existence requires a geometric context.

The distinction here between topology and geometry is the following ill-defined and overly crude dichotomoy. Topology refers to settings (e.g. categories) in which the infinitesimal symmetries (however understood) of the defining structures are infinite-dimensional, whereas geometry refers to setting in which the infinitesimal symmetries are finite-dimensional (maybe infinitesimal is not quite the right word). Proper formulations of such statements require notions like pseudo-groups, sheaves, fiber bundles, prolongations, etc. Put another way, when the defining conditions yield no local invariants, the context is topological, when there are local invariants, the context is geometric. So there is continuous topology, differential topology, symplectic topology, contact topology, etc., in which the symmetries are homeomorphisms, diffeomorphisms, symplectomorphisms, contactomorphisms, etc. and there is always a basic theorem that any two objects in the category are locally equivalent, hence there are no local invariants.

Geometry requires some further structure, generally speaking a connection of some sort. Note that a Laplacian like differential operator is not far from being a connection. In [1] Connes has shown that a smooth compact Riemannian manifold is determined by a spectral triple, which consists of a Hilbert space, an algebra acting on the Hilbert space, and a self-adjoint differential operator (that will be the Dirac operator, a square-root of the Laplacian) that satisfy a series of conditions. In some very rough sense, having the context to make sense of spectral theory determines geometry rather than topology. In one of the other answers there is mentioned the notion of a sympletic spinor. While the bundle of symplectic spinors is defined using what is here being called topological data (spin and metaplectic structures are just refinements of smooth and symplectic structures), the definition of the operator on its sections analogous to the usual Dirac operator requires a connection in some way compatible with the underlying symplectic structure. Thus this notion requires a geometric context too.

From a slightly different point of view, $Sp(n, \mathbb{R})$ is the analogue of $GL(n, \mathbb{R})$, and geometry starts upon passing to a compact subgroup of $Sp(n, \mathbb{R})$, analogous to $O(n)$. The smooth and symplectic frame bundles of a smooth or symplectic manifold are principal bundles for infinite-dimensional groups of diffeomorphisms and symplectomorphisms that contain $GL(n, \mathbb{R})$ and $Sp(n, \mathbb{R})$, but their reductions to $O(n)$ or $U(n)$ bundles are not preserved by the actions of these infinite-dimensional groups.

Unfortunately, because people used to view a symplectic form as an antisymmetric analogue of a Riemannian metric, what here is called "symplectic topology" is often called "symplectic geometry". To my mind this is misleading because it is like calling "differential topology" by the name "differential geometry". When one speaks of differential geometry, one generally has in mind a smooth manifold equipped with a metric. Similarly, symplectic geometry ought to mean a symplectic manifold equipped with some further geometric structure (the most direct analogues with the smooth setting being Hermitian or Kähler metrics; the issue of integrability means that there is no longer a single analogue, rather several).

Not everyone will agree with the terminological distinction being made here nor with how it is made, and I have intentionally avoided making it precise, but it seems to me the key is the distinction between infinite-dimensional and finite-dimensional "symmetries", however one chooses to make it precise.

I don't know that the premise is correct. It seems that there is symplectic spectral geometry. See, for example:

Vassilevich, Dmitri, Spectral geometry of symplectic spinors, J. Math. Phys. 56, No. 10, 103511, 10 p. (2015). ZBL1325.81105.