In one variable, a polynomial (of any degree) is determined by its values on a finite set of points. More specifically if $p$ is a polynomial of degree $k$, and $x_0 , \dots x_{k}$ are points for which we know the values $\{p(x_{i})\}_{i=0}^{k}$, then we can determine $p$. There are several ways of doing this, but the most elementary is probably by setting
$$Q_i(t):=\prod_{\substack{0\leq j\leq k}\\ \:\:\:j\neq i}\frac{t-x_j}{x_i-x_j}$$ and then it's easy to show that $$p=\sum_{i=0}^{k}p(x_i)Q_i.$$
I want to know if the same is true for polynomials in multiple variables or if there is at least a similar statement. I imagine in multiple variables, polynomials are determined (at the very least) by their values on a full rank lattice in $\mathbb{R}^n$ (e.g., $\mathbb{Z}^n$), though I suppose I don't actually have a proof of that. It just seems highly likely. Theorems, proofs, and references much appreciated.
Yes, the multivariate polynomial is uniquely determined by its values on the lattice points.
Reference to a stronger result: Alon, N., Tarsi, M. Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992). https://doi.org/10.1007/BF01204715 , lemma 2.1
(based on an answer of @LinAlgMan to this question Vanishing of a multivariable polynomial on a lattice)