This is a great question. (I am the first author on the "Floquet time crystals" paper referred to btw). The analogy is not perfect because what actually happens

This is a great question. (I am the first author on the "Floquet time crystals" paper referred to btw). The analogy is not perfect because what actually happens to a spin-1/2 particle under a 360 degree rotation is it comes back to the same configuration, but its wavefunction picks up a quantum phase factor of (-1), which is not observable. On the other hand, a Floquet time crystal does actually does go to an observably different state under a time shift. The best analogy is really to, for example, a magnet, which does go to an observably different state if you rotate it (because the north and south poles rotate).