This is a special case of this question.

Let $\mathbb{F}$ be a finite field and $\mathbb{F}_{\leq d}[x,y]$ the set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Do there exist non-empty, disjoint subsets $A,B\subset\mathbb{F}_{\leq d}[x,y]$ such that for all pairs $(\ell,P)$, where $\ell\subset\mathbb{F}^2$ is a line, and $P$ is a univariate polynomial of degree at most $d$ defined on $\ell$ we have: $$\Big|\{Q\in A:Q|_\ell=P\}\Big|=\Big|\{Q\in B:Q|_\ell=P\}\Big|.$$

**Follow-up Question:**
Do there exist $A,B\subset\mathbb{F}_{\leq d}[x,y]$ satisfying the above and also $|A|=|B|=|\mathbb{F}|^{\mathcal{O}(1)}$?