What is the physical meaning of the von Kármán constant?
K is in fact just an empirical proportionality value without any significant meaning. In fact each of the tracers (such as temperature, humidity and ...) should have had their own K BUT we dump that coefficient into the (star)* version of that tracer, for example T* (near surface temperature) or Hum* (near surface humidity), so we can still use K in their respective formulation.

Monin-Obukhov length has K in it because that length is related to u*. That length comes from dividing the Shear production and buoyancy procution/dissipation term in turbulence kinetic energy (TKE) equation. As you know Shear production is a function of momentum transfer and u*. Since there is a u* there is a k in there too. 

Unlike other stuff in turbulence, the story of K is unfortunately boring.