In the $3D$ plane, how is $z$ always the "vertical" axis? In all the graphs I see of the 3d cartesian system, $x$ and $y$ are "horizontal", although they can change positions, but $z$ is always the vertical one.

Even if we had a choice to graph the $yz$ plane, the $z$ will always be the vertical axis, in almost every textbook and example. Why is this?

Is this convention?

I even see in a few youtube videos that $z$ is always the vertical axis, no matter what. How come this is the case?
Besides as you said it is convention, the direction of the $z$-axis can be determined by using the "Right Hand Rule". When we take unit vectors $\hat{i}$ and $\hat{j}$ from $x$-axis and $y$-axis respectively, we have $\hat{i} \times \hat{j} = \hat{k}$, where $\hat{k}$ is the unit vector on $z$-axis. Since the direction of cross-product will be determined by the Right Hand Rule, whether it is vertical or not is all about how the directions for $x$-axis and $y$-axis are determined (And since they generally are lying on a horizontal plane, your result follows).
It is the same reason why we call some cars as righthand drive, one could have equally well have called it rigthfoot drive (we use a our foot a lot to drive)