Let $P = [0,m] \subset M_{\mathbb{R}^2}$ be the line segment and consider the cone over $P$.
What is the toric variety of the cone over $P$?
The thing is I am not entirely sure how to construct a cone $\sigma$ from the polytope $P$. Should it just be the cone generated by $me_1$? In which case wouldn't we obtain the variety $\mathbb{C} \times T^1$, where $T$ denotes the torus?
What does the "affine variety of the cone of a polytope" actually mean? Sorry if this question looks too trivial, but I couldn't find any reference that answers this question explicitly.
First there is two ways to get a toric varieties : via fan or via polytopes. It seems you are considering the polytope version here, which looks strange because you took a 1-dimensional polytope in $\Bbb R^2$ which is a degenerate case. You can just look at $P \subset \Bbb R$ and you will get a projective line (but remember than a polytope is a bit more than simply a toric variety, it also gives you a divisor. In this case your segment correspond to $\mathcal O(m)$.
Here is a possible interpretation in general : suppose $P \subset M$ is your polytope. Embed $M$ in $M \times \Bbb R$ as $M \times {(0,1)}$ and take the cone over $P$ with respect to the origin. This gives you a polytope hence a toric variety.
P.S : for a segment, the associated toric variety is $\Bbb P^1, \Bbb C, \Bbb C^*$ depending if the segment is closed, half-closed or open.