Consider a smooth closed manifold $M$. The space of Riemannian metrics is an open cone in the space of sections of some vector bundle. On this space the group of diffeomorphisms of $M$ acts by pullback. The stabilizer of this action at a given $g$ the stabilizer is exactly the isometry group of $(M,g)$: This is a finite dimensional compact Lie group.

I am interested in the quotient space. Let's agree to call this space the space of Riemannian structures. This space parametrizes the metrics on $M$ up to isometry.

As the action of the diffeomorphism group is not free, I do not expect this to carry a natural manifold structure.

What type of space is the space of Riemannian structures?

To me this really feels like an infinite dimensional version of an effective orbifold: These spaces are exactly quotients of smooth manifolds acted upon by Lie groups with discrete stabilizers. But in this setting everything is shifted up "in dimensionality": discrete $\rightarrow$ finite dimensional $\rightarrow$ infinite dimensional.

Is there such a theory of "orbifolds", modelled on infinite dimensional manifolds quotiented by compact Lie group actions?