The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_{un})$ over the field with one element $\mathbb{F}_{un}$ is, simply by taking $q=1$, $[n]_1=n$. Similarly, whenever the number of $\mathbb{F}_q$ points on a variety is a polynomial in $q$, we can take the $q=1$ limit just as well.

On the flip side of the coin, the $q$-integers appear in many $q$-analogs, including representation theory of quantum groups. In this context, $q$ is often a root of unity $q=e^{\frac{2\pi i}{k}}$ or more generally a complex number, and $q\rightarrow 1$ limit is understood as a "classical limit".

The connection is probably not superficial, because there are other instances where $q$ is interpreted in two different ways. For instance, according to this slide, it's a theorem of Katz that for a smooth quasi-projective variety $X$ defined over $\mathbb{Z}$, if the number of $\mathbb{F}_q$ points is a polynomial in $q$, then it is the E-polynomial of $X$, which is a specialization of the weight polynomial $WH(X;q,t)$. On the other hand, Chuang-Diaconescu-Pan related the weight polynomial to the refined Gopakumar-Vafa expansion.

So, my question is, is there any simple explanation of the apperance of $q$ in two different guises, a power of a prime and a root of unity?