For a rig (or semiring) $R$ we can define for $n \in \mathbb{N}$ an element $n * x = \underbrace{x + \dots + x}_n$.
Is there a standard name for the property that $n * x = n * y$ implies $x = y$ for any $n \geq 1$ and $x, y \in R$?
Or does it follow from any standard property?
Personally, the name I have encountered this under is "torsionfree ring $R$" meaning that the underlying abelian group is torsionfree: $nx\neq 0$ if $n\in\mathbb N\setminus \{0\}$ and $x\in R\setminus\{0\}$.
Another way to say it is that there are no nonzero elements of $R$ that have finite additive order.
This is equivalent to what you said in the case of rings if you group what you wrote as $n(x-y)=0$, and follow the consequences. But it is only a consequence of your definition for semirings.