What is the rule for something divided by itself equaling 1?

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.

If $0/0 = x$ then $ 0=0x$ which is true for any value of $x$, meaning there is no unique solution: $x$ is undefined for this kind of ratio.

So then y/y is undefined unless it is specified that y≠0