cos(πn/3) with frequency increased by 2π

$x[n] = \cos\left(\frac{\pi n}{3}\right)$ has $\omega_0 = \pi/3$ and period $N = 6$. Increase the frequency by $2\pi$:

$$\begin{aligned} y[n] &= \cos\left[\left(\frac{\pi}{3} + 2\pi\right)n\right] = \cos\left(\frac{\pi n}{3} + 2\pi n\right) \\ &= \cos\left(\frac{\pi n}{3}\right) = x[n] \end{aligned}$$

because $2\pi n$ is an integer multiple of $2\pi$ for every integer $n$.

Nothing changes: the new signal is identical to the original, with the same samples and the same period $N = 6$. DT sinusoids whose frequencies differ by $2\pi$ (or any multiple of $2\pi$) are the same signal. So the frequency of a DT signal is unique only over an interval of length $2\pi$ (e.g. $-\pi < \omega \le \pi$ or $0 \le \omega < 2\pi$), unlike CT signals, where a higher frequency always oscillates faster. This is the basis of aliasing in sampling.