Consider the product $\mathbb{A}^n \times \mathbb{P}^{m}$. Let $x_i$ be affine coordinates on $\mathbb{A}^n$ and $y_j$ homogeneous coordinates on $\mathbb{P}^{m}$.
Question: Is $A=k[x_1,\dots,x_n,y_0,\dots,y_m]$ the coordinate ring of $\mathbb{A}^n \times \mathbb{P}^{m}$ "in some sense"? If yes, how can we rigorously see that and what is "that sense"?
On
Yes. $X=\mathbb A^n \times \mathbb P^m$ is a smooth toric variety, and thus have a homogeneous coordinate ring (in the sense of Cox). It is graded by the Chow group $A_{n+m-1}(X)$, and homogeneous ideals (w.r.t. this grading) correspond to subvarieties on $X$ just as in the projective case.
See the description here.
Your intuition is correct, this should be the coordinate ring in some sense and it is. You just need to be careful to treat the $x$ variables and the $y$ variables somewhat differently. In particular you want to treat the $y$ variables as homogeneous variables and the $x$ variables as regular affine ones.
To do this in one step: Define a grading on this ring by putting the $x$ variables in degree 0, and the $y$ variables in degree 1. Now if you take $\textbf{proj}$ with respect to this grading you get what you want.