The Fourier transform
and welcome!
You're right that the Fourier transform is just a mathematical operation that you can perform on any function. You can perform a Fourier transform in space (take the Fourier transform of an image, for example), on voltages as a function of time, take the fourier transform of some differential equation, etc.
The reason why \$\omega\$ corresponds to the angular frequency when applied to a signal in the time domain comes from the details of the math. The Fourier transform for a signal \$f(t)\$ can be written as:
$$
F(\omega) = \int f(t) e^{-i\omega t}\, dt 
$$
so what does that mean? Well, you can write \$ e^{-i\omega t}= \cos(\omega t)-i\sin(\omega  t)\$ (Euler's rule), so then you start to see clearly what \$\omega\$ means.
When \$\omega t = 2 \pi\$, then \$\cos\$ and \$ sin\$ would have gone one full cycle. So the amount of time for one full cycle (the period) is:
$$
T_P = \frac{2 \pi}{\omega}
$$
So that's the number of seconds per cycle. The number of cycles per second (the frequency in Hz) can then be found:
$$
f = \frac{1}{T_P}=\frac{\omega}{2\pi}
$$
So that's why the \$\omega\$ of the Fourier transform of a signal is the angular frequency (\$\omega\$). Which is related to the frequency in Hz using the equation above.