How is frequency defined for a periodic signal?
In my world, frequency is inverse to period, i.e., a sinusoid with fundamental period $T$ has a frequency $f = \frac 1T$. Of course, angular frequencies are also common, in which case we have $\omega = 2\pi f = \frac{2\pi}{T}$, I guess this is what you were referring to.

If a signal is periodic but not purely sinusoidal, it does not have "just one" frequency. Instead, it will have a line spectrum, i.e., frequency content at integer multiples of its fundamental frequency. The fundamental frequency is, as above $\frac 1T$ if $T$ is the fundamental period, $\frac{2\pi}{T}$ if you prefer angular frequencies.

To be a bit more precise, if your signal $x(t)$ is periodic with period $T$ so that $x(t) = x(t+T) \; \forall t$, you can expand it into its Fourier series given by $$x(t) = \sum_{\ell = -\infty}^\infty \gamma_\ell {\rm e}^{\jmath \omega_0 \ell t},$$ where $\omega_0 = \frac{2\pi}{T}$ is the fundamental frequency and the coefficients $\gamma_\ell$ tell you how much energy there is in the different overtones of the fundamental.

*edit: Regarding your request for clarification "As there's "a time-period of a signal", there's no such thing as "a frequency of a signal"?"
You might think so, but I guess that's not the answer here. See, if a signal has fundamental period $T$, it is also periodic with $2T$, $3T$, $4T$ and to so. This means while we could consider $\frac 1T$ its base frequency, we might also consider $\frac{1}{2T}$, $\frac{1}{3T}$, $\frac{1}{4T}$ and so on, right? But here is the thing: this is also true for the pure sinusoid. And yet, its spectrum only has a line at $\pm \frac{1}{T}$. Which is the base frequency if you consider the base period, it would be the first overtone if you were to consider twice the base period. 

So, that's not what's happening. What's happening is that the spectrum tells us how similar the signal looks to harmonics of certain frequencies. If you're given a harmonic, you just need one to explain the signal. If you're given a non-harmonic but periodic signal, one harmonic is not enough, you'll need the overtones as well.