Recovering DTFT from Z-transform
The $\omega$ in the frequency response of a discrete-time system $H(e^{j\omega})$ is indeed unitless. The frequency response $H(e^{j\omega})$ is periodic with period $2\pi$. If the discrete signal is obtained via sampling with sampling rate $f_s$, then the relation between $\omega$, the actual frequency $f$, and the sampling frequency $f_s$ is
$$\omega=2\pi\frac{f}{f_s}\tag{1}$$
Consequently, as a function of $f$, $H(e^{j2\pi f/f_s})$ is periodic with the sampling frequency $f_s$.