What is the physical significance of damping coefficient?

The damping coefficient applies both to hydraulic circuits and springs.

Hydraulics

In general, higher the coefficient or viscosity higher is the tendency to ensure steady flow and hence a higher system efficiency. This is governed by the properties of hydraulic oil selected for use.

Springs

Higher the coefficient, greater is the tendency for a spring to reach a desired equilibrium position. This is governed by the properties of spring material selected for the applciation.