Two different multiplications of the equations does not give the same answer.

The product of $y=x^2$ and $y=x^3$ is $y^2=x^5$.

By manipulating the equations to $0=x^2-y$ and $0=x^3-y$ first their product is $0=(x^2-y)(x^3-y)$ or $0=y^2-yx^2-yx^3+x^5$.

Please explain why their roots are different!!!

Part of the problem is the meaning of the equals sign “$=$”. It means “is”. It means that the expressions on the left and right of the equals sign are descriptions of the same one number, signify the same one number.

You want to write $y=x^2$ and you want to use another equation $y=x^3$. If both instances of the $y$ and the $x$ refer to the same number, then it makes sense to “multiply” the two equations, even though there might be little reason to do so. If the two $x$’s, call them $x_1$ and $x_2$, are not the same, then you’d have to keep the subscripts when multiplying.

But when you take the two equations together, I look at them and say that if we’re accepting both as describing the same $x$ and the same $y$, then certainly $x^2=x^3$, true only of $x=0$ and $x=1$.