If $f(X,Y|Z)=f(X,Y)$, then is true that $f(Y|X,Z)=f(Y|X)$? Assume that $W = (X,Y)$, $Z$ is some random variable not in $W$, and we know that
$$
f(W|Z)=f(W).
$$
Is it then also true that
$$
f(Y|X,Z) = f(Y|X)?
$$

It's true. Making the notation is a bit more clear:
$$f_{Y|X,Z}(y|x,z)={f_{X,Y|Z}(x,y|z)\over f_{X|Z}(x|z)}={f_{X,Y}(x,y)\over f_{X|Z}(x|z)}=\frac{f_{X,Y}(x,y)}{f_X(x)}=f_{Y|X}(y|x)$$
Since $W$ is independent of $Z$, each part of $W$, i.e. $X,Y$ should be independent  of $Z$. This yields $f_{X|Z}(x|z)=f_X(x)$.
