We study a topological superconductor island in the Coulomb blockade regime, hosting spatially separated Majorana modes coupled to multiple external normal leads via single electron tunneling. We show that at low temperatures the system flows to a non-Fermi liquid fixed point and displays a topological Kondo state. Remarkably,  this state is stable against tunnel couplings anisotropy or detuning away from the charge-degeneracy point. As a result, the system exhibits a universal conductance at zero temperature, which is a fraction of the conductance quantum, and low-temperature corrections with a universal power-law exponent. In addition, we show that coupling the island to a bulk superconductor leads to instability of Kondo state and to a new non-Fermi liquid fixed point of Andreev reflection.