We consider an annular superconductor-insulator-superconductor Josephson-junction, with the insulator being a double layer of electron and holes at fractional quantum Hall state of identical fillings. When the two superconductors gap out the edge modes the system has a topological ground state degeneracy in the thermodynamic limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional limit, where the width of the insulator becomes small, the ground state energies are split. We discuss several implications of the topological degeneracy that survive the crossover to the quasi-one-dimensional limit. In particular, the Josephson effect shows a 2πd-periodicity, where d is the ground state degeneracy in the 2D limit. We find that at special values of the relative phase between the two superconductors there are protected crossing points in which the degeneracy is not completely lifted. These features occur also when the insulator is a fractional topological insulator. We describe the latter using a wires construction. Furthermore, when the superconductors are replaced by ferromagnet the Josephson effect is substituted by a spin Josephson effect.