I agree that Joyal's species are sort of key here, although the kinds of data types being considered here are not exactly the same thing.
My guess is that 1/(1-A) should be seen as a kind of "generating function" for the formal power series 1 + A + A2 + A3 ... and that the relevance of power series can be explained by some simplicial structure hiding somewhere. It's certainly true that one can take derivatives and integrals of these data types: see Conor McBride, The Derivative of a Regular Type is its Type of One-Hole Contexts.
There is a direct interpretation of division which comes up when computing "Taylor series" of these datatypes: e.g. the type An/n! is the n-element multiset, since An is an n-tuple and n! can be read as the action of the permutation group. Unfortunately, the formalism does not quite work out: see http://www.cs.nott.ac.uk/~ctm/Dissect.pdf for a reference.