Here’s a good document defending the merits of this design. https://xenaproject.wordpress.com/2020/07/05/division-by-zer...

Doesn't this allow one to prove x=y for any x, y?

x/0 = x(1/0) = x*0 = 0, so x/0 = 0 for all x.

So x/0 = y/0.

Multiply both sides by 0: x = y.

What theorem did you use that allowed you to multiply both sides by $0$? (That theorem had conditions on it which you didn't satisfy.)

No, because x/y is just an arbitrary operation between x and y. Here you're assuming that 1/x is the inverse of x under *, but it's not.

I mean in a normal math curriculum you would define only the multiplicative inverse and then there is a separate way to define fraction, if you start out with certain rings. It is kind of surprising to me that they did a lazy definition of division.