What is stationary discontinuity how is it different from contacts and shocks

A stationary shock is one example of a stationary discontinuity. Since shocks and contact surfaces can be moving through a fluid, the 'stationary' descriptor is prepended to denote the type of discontinuity being referenced. This also applies to other types of discontinuities.

Since the classical laws of physics must be Galilean invariant, we can use any uniformly moving reference frame to describe the physical phenomenon. How does one define a stationary discontinuity in that case ?

Stationary is a name then.

What about if this discontinuity is a contact wave i.e. one eigenvalue for a system of conservation laws is it still a stationary discontinuity then?

Generally no. The eigenvalue typically corresponds to the speed of the discontinuity.