In what sense is the wave motion of a guitar string analogous to the motion of an electron in an atom?

A guitar string vibrates when we strikes it. It starts vibrating in several modes simultaneously. It stretches between the saddle and the nut. This distance represents one-half wavelength.

Now if we consider that the string forms a circle, then we have an interpretation of an electron which vibrates in the orbit surrounding the nucleus. We are aware that electrons have wavelength. If the circumference of the orbit happens to be the integer multiple of wavelength, then the orbit is "allowed" since "the electron will retraces its own path."

This explains the line spectrum and not a continuous spectrum.

A line spectrum refers an electron that jumps between the specific energy levels, thus producing only specific colors.