Zn- and O-isotopically enriched ZnO nanorods were grown with excellent optical quality allowing an identification and resolution of various bound exciton zero-phonon lines (ZPL). Furthermore, the well-known Cu-related emission at 2.86 eV could specifically be studied in order to investigate the local environment of the defect including possible involvement of native defects such as interstitials and vacancies in this deep centre. Energetic shifts of this ZPL were measured and compared to changes in the near band edge (NBE) energies as a function enrichment. No relative shift was observed in Zn-enriched samples, indicating that only O atoms lie in the immediate vicinity of the Cu atom, and that Zn interstitials and O vacancies are not involved in this defect. NBE and Cu 2.86 eV ZPL emissions in samples with O enrichment displayed a significant relative shift, and the Cu 2.86 eV ZPL line widths showed a substantial increase, which is attributed to the multiple local configurations possible for O atoms surrounding the Cu atom in these mixed isotope environments. These data provide the first direct evidence of the microscopic nature of this defect centre of a Cu atom substituting on a Zn lattice site, and is thus consistent with the conventional model of this defect.