Can we consider $\omega$ (angular frequency) in equation of simple harmonic motion (SHM) as the angular velocity of the object in circular motion, when we see simple harmonic motion as projection of circular motion?
Yes. The point on the circle with radius $R$ revolves around the centre, with angular velocity:
The projection on the RHS axis, call it $y$, is:
where $\phi$ is the angle $\theta$ at $t=0$.