The $q$-analog $[n]$ of a whole number $n$ is $q^{n-1}+\cdots+q+1$, in which case the binomial and $q$-binomial are
$$ \binom{n}{k}=\frac{n(n-1)\cdots}{k(k-1)\cdots} \qquad \left[\begin{matrix} n \\ k \end{matrix}\right]=\frac{[n][n-1]\cdots}{[k][k-1]\cdots} $$
Then $\binom{n}{k}$ counts the $k$-subsets of $\{1,\cdots,n\}$ and $\left[\begin{smallmatrix}n \\ k \end{smallmatrix}\right]$ counts the $k$-subspaces of $\mathbb{F}_q^n$.
Further, $\left(\!\binom{n}{k}\!\right)$ ("$n$ multichoose $k$") counts how many $k$-submultisets of $\{1,\cdots,n\}$ there are (that is, multisets containing $k$ of the numbers $1$-$n$ counted with multiplicity). The standard stars-and-bars argument shows $\left(\!\binom{n}{k}\!\right)=\binom{k+n-1}{k}$.
Question. Does $\left[\!\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]\!\right]:=\big[\begin{smallmatrix}k+n-1 \\ k\end{smallmatrix}\big]$ count something geometric / linear-algebraic associated to $\mathbb{F}_q^n$?
Ideally, such an interpretation leads to enriched analogies.
We have $\dim\Lambda^k\mathbb{F}^n=\binom{n}{k}$ and $\dim S^k\mathbb{F}^n=\left(\!\binom{n}{k}\!\right)$, and moreover $|\mathrm{Gr}_k(\mathbb{F}_q^n)|=\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]$ so perhaps $\left[\!\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]\!\right]$ counts the elements of some variety within $S^k\mathbb{F}_q^n$. Indeed, the collection of $k$-subsets of $\{1,\cdots,n\}$ form an $S_n$-set equivalent to $S_n/(S_k\times S_{n-k})$, and the collection of $k$-multisets of $\{1,\cdots,n\}$ is equivalent as an $S_n$-set to $\bigsqcup S_n/(S_{u_0}\times\cdots\times S_{u_n})$ where the disjoint union is taken over solutions to the system
$$ \begin{cases} 0u_0+1u_1+2u_2+\cdots+nu_n=k \\ ~u_0~+~u_1~+~u_2~+\cdots+~u_n=m \end{cases} $$
(Interpret $u_m$ as saying how many elements of $\{1,\cdots,n\}$ have multiplicity $m$ in the multiset.)
Similarly, $\mathrm{Gr}_k(\mathbb{F}_q^n)\cong \mathrm{GL}_n(\mathbb{F}_q)/B$ where $B$ is the subgroup of block upper-triangular matrices, with $k\times k$ and $(n-k)\times(n-k)$ blocks. Or, $SO(n)/\big(SO(k)\times SO(n-k)\big)$ if $\mathbb{F}=\mathbb{R}$. Perhaps the variety in $S^k\mathbb{F}_q^n$ can be decomposed in a similar way to the collection of $k$-multisets of $\{1,\cdots,n\}$, if it indeed exists.
One tricky thing here is that $\left(\!\binom{n}{k}\!\right)$ and $\binom{k+n-1}{k}$ do not represent equivalent $S_n$-sets, the latter decomposing into orbits according to the Vandermonde convolution identity, so presumably something similar can be said about $\left[\!\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]\!\right]$ and $\big[\begin{smallmatrix}k+n-1 \\ k\end{smallmatrix}\big]$ representing different things (in terms of either group actions or representations of $GL_n\mathbb{F}_q$).
Moreover, $\left[\begin{smallmatrix}n \\ k \end{smallmatrix}\right]$ is a generating function for the Schubert cells of the Grassmannian $\mathrm{Gr}_k\mathbb{F}^n$ (corresponding to RREFs of $k\times n$ matrices) and their dimensions, so perhaps something similar is true for $\left[\!\left[\begin{smallmatrix} n \\ k \end{smallmatrix}\right]\!\right]$ and the conjectural variety in $S^k\mathbb{F}_q^n$.