Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order affine type system with inductive data types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data types which might be defined by mutual/nested recursion. We show how to achieve this for all types by taking models of a first-order linear type system whose atomic types are discardable and then presenting an additional affine interpretation of types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data types.