Very basic question about how we define frequency in signal processing

When talking about general periodic continuous-time signals for which

$$x(t + T_0) = x(t)$$ where $T_0$ is the fundamental period we define the fundamental frequency $\omega_0$ as $\omega_0 = 2\pi/T_0$.

The way I interpret this is that $1/T_0$ is the frequency of the signal in cycles per second and there are $2\pi$ radians in one cycle, therefore the angular frequency is $2\pi/T_0$ radians per second. But are there really $2\pi$ radians in one cycle in general?

Take $x(t) = \tan(t)$ for example. Its fundamental period is $\pi$ and using the definition of fundamental frequency above, its fundamental frequency is 2 radians per second. But in the case of the tangent function, aren't there $\pi$ radians in one cycle, as opposed to $2\pi$ radians? This would give us a definition of frequency as $\omega_0 = \pi / T_0$. Or do we refer to the unit circle by convention when talking about "cycles"?

Apologies if this question sounds very basic.