The orientation of the axes is conventional.
If you just say $x,$ $y,$ and $z$ with no indication that you might be
switching things around, then people will generally assume
something like your figure on the right, where the axes are arranged
according to the right-hand rule.
But a left-hand-rule set of axes is perfectly OK if you say you are using
a left-hand rule (or sometimes even when you don't say so, for example
if you say the axes can be any orthogonal set).
Likewise it is conventional for a Cartesian plane to have
$x$ and $y$ coordinates, for the $y$ axis to be plotted vertically,
and to plot functions in the form $y=f(x)$.
But you can plot $x=g(y),$ you can label the axes $x$ and $z$ or $x$ and $t$
or even $u$ and $v$, you can put the $x$ axis vertical, and so forth.
You will just have to be more careful to be sure that people understand
which coordinate system you are using and how you are using it.
(It may help if there is a reason why you are doing in that way, too.)
If you take something like $y=f(x),$ which you know how to plot in the
conventionally-labeled Cartesian plane, and do something unusual such as
swapping the $x$ and $y$ axes, you end up with a graph that looks something
like what you had before, but maybe flipped and/or rotated (and for more
complex changes of coordinates, for example non-orthogonal ones, maybe
stretched and/or skewed).
In particular, the graph of $y=f(x)$ plotted on a graph with
$x$ and $y$ axes swapped will look just like the normal graph if you rotate the
paper so the $y$ axis is vertical again and look at it in a mirror.
A real-life example of different uses of coordinates occurs in aeronautical
engineering, where it is common to attach body-centric $x$, $y$, and $z$
coordinates to an aircraft with the $z$ coordinate pointing down.
This can be confusing if you have not seen it before, but in that particular
application it is common enough to be its own convention.