Recent experiments have shown that self-propelled millimetric walking droplets bouncing on a vibrating liquid surface exhibit phenomena, such as interference or tunneling, that so far were thought to be possible only in the microscopic realm. Here we present calculations showing that the surface wave satisfies, in the long-memory limit, a Bohr-Sommerfeld quantization-like relation. This strongly suggest the possibility of a novel fundamental type of quantization in these experiments, which can simultaneously explain their emulation of the quantum behavior and, more importantly, shed light into some of the interpretational difficulties of the standard quantum theory.