Remember, the conditional probability is a function, not a number, so you need to be careful when manipulating these quantities. I will use the same interpretation of the question as David Moews did in his answer that you linked to. Thus, $W,X,Y,Z$ are random variables defined on the same finite probability space.
So the notation $\mathbb P(X,Y\mid W,Z)$ is shorthand for: the function $f(w,x,y,z)$ given by
$$
f(w,x,y,z)=\mathbb P(X=x,Y=y\mid W=w, Z=z)=\frac{\mathbb P(W=w,X=x,Y=y,Z=z)}{\mathbb P(W=w,Z=z)},
$$
where $w$ and $z$ range over all values such that $\mathbb P(W=w,Z=z)>0$.
Similarly, the notation $\mathbb P(X,Y,Z\mid W)$ is shorthand for the function
$$
\frac{\mathbb P(W=w,X=x,Y=y,Z=z)}{\mathbb P(W=w)},
$$
with a similar caveat on the value of $w$. (I will stop mentioning this for the rest of this posting.)
Continuing in this way, we can write the given condition
$$
\mathbb P(X,Y,Z\mid W)=\mathbb P(X\mid W)\mathbb P(Y,Z\mid W)
$$
explicitly as follows:
$$
\frac{\mathbb P(W=w,X=x,Y=y,Z=z)}{\mathbb P(W=w)}=\frac{\mathbb P(W=w,X=x)}{\mathbb P(W=w)}\frac{\mathbb P(W=w,Y=y,Z=z)}{\mathbb P(W=w)}.
$$
Summing both sides over all $y$ yields
$$
\frac{\mathbb P(W=w,X=x,Z=z)}{\mathbb P(W=w)}=\frac{\mathbb P(W=w,X=x)}{\mathbb P(W=w)}\frac{\mathbb P(W=w,Z=z)}{\mathbb P(W=w)}.\qquad (1)
$$
Using this equation, I will show that your last equation (which you marked with an inequality) is in fact true, resolving the confusion. That is, I will show that the previous equation implies
$$
\mathbb P(X\mid W)=\mathbb P(X\mid Z,W)
$$
or equivalently in long form
$$
\frac{\mathbb P(W=w,X=x)}{\mathbb P(W=w)}=\frac{\mathbb P(W=w,X=x,Z=z)}{\mathbb P(W=w,Z=z)}.\qquad (2)
$$
So how do we show that $(1)$ implies $(2)$? Well, remember that on the right hand side of $(2)$, $w$ and $z$ take values such that $\mathbb P(W=w,Z=z)>0$. Thus, we can apply $(1)$ to $w,z$ with this property, in which case it is legal to divide through on both sides by $\mathbb P(W=w,Z=z)$, and this takes us from $(1)$ to $(2)$, after cancelling out the repeated factor of $\mathbb P(W=x)$ from both sides.
