Just from a coordinate standpoint, without regard to application.. if the $x$- and $x’$-axes are coincident and the other axes are parallel, then $y=y’$, $z=z’$. It’s true that the infinite planes $x=0$ and $x’=0$ would cover the same points, but the points would have different values for the other two coordinates, and not be the same points (ie different coordinates). The set-up is: any point $(a, b, c)$ in $XYZ$ would equal $(a, b+j, c+k)$ in $X’Y’Z’$, where $j,k$ are constants.
So it’s not generally true even before looking at the application that $y=y’$.
For the initial set-up, just take a single $x$ axis, and that passes (perpendicularly of course) through the $yz$-plane at a differently numbered point than on the $y’z’$ plane. Those two planes are otherwise coincident, ie are coincident. The $xy$ and $x’y’$ planes are parallel, as are $xz$ and $x’z’$.
If it is moving, we allow $j,k$ to change. Even before any physics, we know they are not moving relatively in the $x$ direction. From a special relativity standpoint this means that the coordinate systems could be moving relatively on $y$ and/or $z$. And $v=\sqrt{\frac{d y}{dt}^2 +\frac{d z}{dt}^2}$ according to both of them. They will not measure objects as having the same length in those directions and will have time dilation if one is faster from a still frame.