What is discrete angular frequency?
$\omega$ is angular frequency in radians, $\omega =2\pi fT$, where $f$ is frequency (for example in Hz) and $T$ is sampling period (for example in seconds). This is revealed by Eq. 3:


  $$Y(\omega, k) = \textstyle\sum_{n=-\infty}^\infty y(n)w(k-n)e^{-j\omega n}\tag{3},$$


which represents discrete-time Fourier transform (DTFT) of windowed signal $y(n)w(k-n)$.

It's not common to call $\omega$ "discrete angular frequency index", which gives just 3 google hits.