In classical mechanics, you can just say that there are n electrons, each with a coordinate in the 7-dimensional space. In the continuum limit, you end up with a phase space density f which evolves in 7 dimensions -no more are needed. You're right that the number of degrees of freedom is still larger than 7, but they all commute and in terms of how you do physics you are basically using 7 dimensions.
In principle the Gibbs measure is defined on top of the full configuration space. For free particle, you can factorise joint probabilities and everything is in principle characterised by one probability distribution on phase space. But this is only true for non-interacting particles! When you consider systems with interaction, correlation functions will be no longer negligible and joint probability distributions do not factorise. Of course, you can compute marginals by computing all but one particle and find a probability distribution on your single-particle phase space, but this is not the full story.
In quantum systems, you have exactly the same story. If you consider a system of free particles, you will find that your thermal states are quasi-free states. This means that they are characterised by the single-particle covariance matrix and that expectation values for many-particle observables factorise. Note that in bosonic systems, these guys give you Wigner functions which are Gaussians.
Once you consider interacting systems, you will have correlations in your KMS states, and, indeed, expectation values of many-particle observables will not factorise. But this is hardly different from the classical case. You can still integrate (or trace) out all but one particle and study the marginal distribution on single-particle phase space.
This does not mean, of course, that quantum and classical states are the same, they certainly are not. However, when we are merely discussing these dimensionality issues, there is really not that much of a difference.
I think you are comparing very specific classical states to a very broad set of quantum states.