Why I can't divide by y in this equation: 4y = y?

You can't divide by $0$. Rigorously, there are two cases to consider:

Case 1: $y=0$

This is seen to be a solution.

Case 2: $y\neq 0$

In this case we can divide by $y$, getting

$$4=1$$

Since this is false regardless of $y$, we get a contradiction, and thus there are no solutions $y\neq 0$ to the equation. So, the only solution is $y=0$.

Transposing

$$ 3y=0,\quad y=0.$$

So division by $ 0$ is forbidden.

Division is better thought of as the multiplication by a multiplicative inverse. All real numbers except zero have multiplicative inverse. So for example in solving $$2y=2$$ you can multiply both sides by the multiplicative inverse of $2$, which is denoted by $1/2$, to get $$\left(\frac{1}{2}\times2\right)y=\frac{1}{2}\times 2$$ $$1\times y=1$$ $$y=1$$ For $0$, you cannot do the same thing because there is no multiplicative inverse. So if you divide both sides by an unknown $y$, you need to assume that it is not $0$. Hence you need to consider the case $y=0$ later as well.