Bode plot graph: Frequency vs. Angular frequency
Transfer functions are commonly specified in terms of \$s = \sigma+j\omega\$, which means the bode plot can be interpreted as the evaluation of magnitude and phase of the transfer function along the (positive) imaginary axis of the \$s\$ plane. The imaginary axis relates to undamped sinusoidal inputs, which are typical signals of interest for system analysis. This direct correspondence is (in my opinion) the reason why bode is typically explained, plotted and taught in terms of angular frequency.

The x-axis and y-axis units and scaling should be taken as any other plot: a matter of personal preference, application relevance and detail representation. Sometimes angular frequency is preferred, other times \$Hz\$. Sometimes logarithm scaling, other times linear scaling. It is still the same transfer function regardless of plotting preferences, so any representation choices, as long as they are sensible, are acceptable.