Yeah, I think this is along the right lines - in the vector space analogy it's like we have a bunch of vectors we can measure (P, T, S, V, etc) but due to the constraints we're actually working in a 2 dimensional space. So we could form a basis from many different choices of vectors, and our coefficients would change accordingly.

As the other commenter said, you can make this analogy rigorous by looking at manifolds (differential geometry). They're a little bit like the non-linear version of a vector space. In this case the set of physically valid values for P, T, S, V forms a two-dimensional surface due to the ideal gas laws, and you can derive local coordinate charts for the surface using any (non-degenerate) pair of these variables.