Binding anyons to optically active excitons offers a route to manipulate fractional excitations and reveal the internal spatial structure of anyons. We establish the formation of anyon bound states between a mobile dipolar exciton and an e/3 quasihole in a twisted bilayer MoTe₂ fractional Chern insulator. Using exact diagonalization of a continuum-based interacting model, we identify a regime of meV-scale binding and strong exciton-quasihole entanglement controlled by exciton mobility and the distance between the exciton and the FCI. Spatial correlations resolve a localized depletion core carrying approximately e/3 charge around the exciton, providing a real-space characterization of the bound state. Beyond its fractional charge, the quasihole exhibits a moiré-induced octupolar structure. Interlayer tunneling anisotropy strongly deforms this structure while leaving the enclosed charge nearly unchanged, linking the internal profile of the anyon to the underlying electronic quantum geometry. These findings establish conditions for robust exciton-anyon binding in moiré heterostructures and show how these composites provide access to properties of fractional excitations beyond their total charge.