I was taught in school that the magnitude of the kinetic frictional force does not depend on the speed.
Hence, the equation of motion of the harmonic oscillator in the presence of friction with the floor would be
$$m\ddot{x}=-kx-f\frac{\dot{x}}{|\dot{x}|}$$
However, the damped harmonic oscillator is modeled with friction proportional to velocity.
$$m\ddot{x}=-kx-c\dot{x}$$
This may consider air resistance to be linearly dependent on velocity, but I don't see why kinetic friction would disappear. Is it assumed that there is no friction with the floor?