Say P is a convex Euclidean polytope, where the origin is not contained in any bounding hyperplane containing a facet of P, with n facets given by f ◦ x = 1 and m vertices v. (That is, each facet equation has been multiplied through as needed to obtain 1 on the right side.) Is it true that the polytope P’ with m facets given by v ◦ x = 1 and n vertices f is combinatorially equivalent to the polar dual P* of P?
Yes. Your polytope $P'$ is exactly the polar $P^*$ of $P$ with respect to the standard unit sphere. Choosing another conic instead of the unit sphere will in general give another polar. See this wikpedia page for the two-dimensional case of polarity with respect to other conics. So your "Cartesian" dual corresponds to the unit sphere.
Polarity/duality of polytopes or more generally, convex bodies, is treated in many books on convexity. Unfortunately, many of them leave a lot of detail to the reader. I recommend Webster's Convexity, Brønsted's An Introduction to Convex Polytopes or Matousek's Lectures on Discrete Geometry.