Aerial robots are increasingly moving from remote observation toward physical interaction with objects, surfaces, structures, loads, and surrounding flows. This review argues that aerial manipulation cannot be understood as classical manipulation simply mounted on a flying base. Because flying agents remain aloft through continuous momentum and energy exchange with the surrounding medium, support, locomotion, stabilization, and task-directed interaction are intrinsically coupled. Building on broad views of manipulation as intentional environmental regulation through physical interaction, we propose a medium-aware interpretation of aerial manipulation in which interaction may be mediated by contact, by the surrounding fluid, or by both. The review organizes biological and robotic examples into a repertoire of interaction modes and a capability ladder, then develops an actuation-geometric viewpoint in which redundancy induces task-equivalent fibers. Internal motion along these fibers can trade energy for active readiness, aerodynamic promptness, and passive medium coupling. This perspective clarifies why aerial manipulation is difficult, why biological flyers remain broader than robotic systems, and how future platforms may command forces while also shaping how the medium acts back on them.