$z_0$ is a theoretical construct that, while useful in its intended uses, cannot be thought of in too much detail as a physical reality. When using a log law to describe wind speed, it represents the distance above the surface at which that log curve decreases to zero. However, if a measurement of speed were made at this height, it would be unlikely to be zero - more detailed, finer-scaled processes dominate here.
(I don't know wind modelling, but by analogy from water I'm guessing a that thin linear boundary layer exists between the ground and the point at which the log curve becomes dominant)
In practice the log-law approach for wind speed is used when dealing with speeds large distances (tens of metres) above the surface, and is not applicable when $z$ approaches $z_0$. Far more detailed techniques would be needed in this realm.