If we agree that $\textbf{(a) }\dfrac{x}{x}=1$, $\textbf{(b) }\dfrac{0}{x}=0$, and that $\textbf{(c) }\dfrac{x}{0}=\infty^{\large\dagger}$, and let us suppose $z=0$:
$$\begin{align*}
z&=0&&\text{given.}\\
\dfrac{z}{z}&=\dfrac{0}{z}&&\text{divide each side by }z.\\
\dfrac{z}{z}&=0&&\text{by }\textbf{(b)}.\\
1&=0&&\text{by }\textbf{(a)}.
\end{align*}$$
Now, from this, we can get say that $\dfrac{0}{0}=1$. What went wrong?
$\dagger$ Or undefined, if you prefer it.
All answers were given, I'll give one answer offering additional information on division by zero.
Actually it's possible to divide by zero. It's usually not possible to divide by zero, because this operation is meaningless(in some contexts). But sometimes, it's useful to be able to divide by zero and there is a theory devoted to it, called Wheel Theory. Have a look at this article too. Just be careful on how do you use this information, there are contexts in which this is useful. At some of them, it's useless.
You cannot divide by zero. So if $z=0$, then you also cannot divide by $z$. Therefore, your second equality $$\frac{z}{z}=\frac0z$$ is ultimately incorrect.
Thou shalt not divide by zero. Your first statement doesn't hold if $x=0$.
First, there are a couple of comments to make.
(a) is only valid for $x \neq 0$; (b) is only vaild for $x \neq 0$ (c) is not defined.
So you don't have the right to divide by $z$ in your second line.
We agree about a little more than what you wrote. We agree that
(a) If $x\neq 0$, then $\frac xx=1$
(b) If $x\neq 0$, then $\frac0x = 0$
We do not agree about $(c)$ at all, we think it is undefined.
Using only (a) and (b), you cannot prove that $\frac00=1.$
You divided by zero on the second line. There are many proofs similar to yours that end up being complete garbage because of division by zero. Although your example doesn't hide it as well.
