From Toric Varieties by Cox, Little, Schenck:
An affine toric variety is an irreducible affine variety $V$ containing a torus $T_N\cong (\mathbb C^*)^n$ as a Zariski open subset such that the action of $T_N$ on itself extends to an algebraic action of $T_N$ on $V$. (By algebraic action, we mean an action $T_N\times V \to V$ given by a morphism.)
Proposition 1.1.11. An ideal $I\subset \mathbb C[x_1,\dots,x_s]$is toric if and only if it is prime and generated by binomials.
The affine variety $V(y) \subset \mathbb C^2$ is isomorphic to $\mathbb C^1$ and we know that $\mathbb C^1$ is an affine toric variety. But the prime ideal $(y) \subset \mathbb C[x,y]$ is not generated by binomials.
Also, $V(y) \cap (\mathbb C^*)^2 = \varnothing$.
Is $V(y) \subset \mathbb C^2$ not an affine toric variety?