Nature of Electromagnetic Waves
You are mixing up two (related) phenomena: EM waves and currents in a conductor, I guess because they both "have a frequency".

EM waves (i.e. perturbations of EM fields that can travel in space) can be generated by time-varying currents flowing in conductors, whereas EM waves hitting conductors can induce currents in them. That's the relationship between the two phenomena. The frequency of the generated EM wave is the same as the current's and vice versa.

This is the general picture, in qualitative terms. Whether or not, in a given case, the generation of EM waves is negligible it depends on the specific problem at hand, and the same is true for whether or not a non-negligible current is induced in a conductor hit by an EM wave.

As a very broad rule of thumb, the radiated emission by a conducting wire becomes relevant when its length is near the wavelength of the wave that would be radiated, which depends on the frequency of the electric signal. The relationship between wavelength \$\lambda\$ and frequency f in vacuum is:

\begin{align*} 
  \lambda=\dfrac{c}{f} 
\end{align*}

where c is the speed of light \$\approx 3\times 10^8 m/s\$.

Therefore a \$f=50\$Hz current would produce an EM wave having a wavelength

\$\lambda=\dfrac{3\times10^8\, m/s}{50\, Hz} = 6\times 10^6 m = 6000\, km\$.

Hence you would need a wire of about that size (6000 km!) to make the radiation of EM waves at that frequency non-negligible.