Is there a name for the mathematical rule/axiom/property $x/x = 1?$
What are the conditions for it to apply?
For instance, the rule does not apply where $x = 0$ or $x = \inf$. I saw one site that claimed it only applied to real numbers, but it does apply to imaginary ones too, so that rule is not complete.
What is the complete rule and what is its name?
Rather than viewing division as an operation in its own right (that would take a dividend and a divisor to a quotient), mathematicians think about inverses of multiplication. So one thinks of $\frac x y$ as $x · y^{-1}$ where $y^{-1}$ is, by definition, a number inverse to $y$, i.e. fulfilling $y·y^{-1} = 1 = y^{-1}·y$.
For example $\frac 3 2$ is rather thought of as “three halfs” (where a half is the inverse of two) than “three divided by two”. (Well, but one still uses the latter parlance, in fact.)
Therefore, I don’t think there is a name for this arithmetic law itself, at least I don’t know of one. But there’s of course one for the concept of inverse elements. A more fundamental concept is that of neutral elements, on which the concept of inverses depend.
And by the way: Division isn’t viewed as an operation by itself because it behaves badly: It is not totally defined as a map, say $ℚ × ℚ → ℚ$ (you can’t divide by zero), it fails to be associative, let alone commutative, and it doesn’t have a neutral element.