Statement - I : A dimensionless quantity may have unit.
Statement - II : Two physical quantities having same dimensions, may have different units.

angle $\theta$is dimension less quantity but has unit radians. Hence statement I is correct.

The dimensional formula of angular velocity $\omega$ and frequency f  are same $T^{-1}$. The unit of angular velocity is radian per sec and the unit of frequency is hertz or per sec. $\mathrm{\omega }=2\mathrm{\pi f}$

Statement II is correct.