Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely related characterization is that this subspace is the kernel of $\Delta^{d+1}$, where $\Delta : \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ is the difference operator defined by $(\Delta P)(x) = P(x) - P(x-1)$. Roughly stated, my question asks for a $q$-analogue of either of these results.

To make this more concrete, use the standard notation, so $[n]_q = 1+q+\cdots + q^{n-1}$, $[n]_q! = [n]_q[n-1]_q\ldots [1]_q$, and $\binom{n}{d}_q = \frac{[n]_q!}{[d]_q![n-d]_q!}$. Fix $N \in \mathbb{N}$ and for each $m \in \mathbb{N}_0$ let

$$u^{(m)}_q = \bigl( \binom{0}{m}_q, \binom{1}{m}_q, \ldots, \binom{N-1}{m}_q \bigr) \in \mathbb{C}[q]^{N}$$

By the first paragraph, for $P \in \mathbb{C}[x]$, we have $( P(0), P(1), \ldots, P(N-1) ) \in \langle u^{(0)}_1, u^{(1)}_1, \ldots, u^{(d)}_1 \rangle$ if and only if $\mathrm{deg} P \le d$. In a current research problem, it's useful that the same holds for $( P(N-1), \ldots, P(1), P(0))$; in fact the evaluation points $0$, $1, \ldots, N-1$ can be varied by an arbitrary affine transformation.

Is there an analogous characterization of the span $\langle u_q^{(0)}, u_q^{(1)}, \ldots, u_q^{(d-1)} \rangle$?

As a follow up, what transformations preserve this space? In particular, is it invariant under a $q$-analogue of the affine transformation just mentioned? Despite some thought I have not found any reasonable answer to these questions, but I find it hard to believe that there is nothing to be said.