We report a microphotoluminescence study of the exciton and the biexciton localized in very elongated islands formed by well-width fluctuations in a thin CdTe/CdMgTe quantum well grown on a vicinal surface. The electron-hole exchange interaction in a local reduced symmetry splits the exciton states. The resulting transitions are linearly polarized along the two orthogonal principal axes of the island. The valence band mixing induced by the elongated shape of the potential leads to a strong polarization anisotropy and to the observation of dark exciton states under magnetic field. The biexciton-exciton transition reproduces all the fine structure of the exciton state including the transition of the biexciton to the dark exciton state.