This seems to be a trivial question but I failed to find a good notation for it. Surely the vertical line passing through $(a,0)$ has the equation $x=a$. But $x=x$ does not seem to be a good equation for the vertical line passing through $(x,0)$.
I have to tag "relations" instead of "functions" as this line does not represent a function.
The problem is that the $x$ you're using for the term "$xy$ plane" (or just the name of the axis) and the $x$ used in the name of the point $(x,0)$ simply have two different meanings, so even attempting to write them like this results in obvious problems.
$x$ in the case of the $x$ axis simply represents a variable, an indeterminant, the input value to your function/relation, whatever you choose to call it.
In the case of $(x,0)$, it is meaning to reference the first coordinate of the pair that the point represents.
The only way to treat these together would be like ... $x$ as a parameterization of the real number line / $x$-axis while in the $xy$ plane, i.e.
$$\text{the x-axis} = \{ (x,0) \mid (x,y) \in \Bbb R^2 \}$$
which is obviously not what you want.
The conclusion being, be very careful with what you name things. In the situation posed, you either get what you don't want or pure nonsense out of using $x$ (or any other such dummy variable) in multiple ways at the same time. Rename the axes, or rename the point. If you decided to work with the $zy$ plane instead, your equation would be $z=x$. If you renamed the point, call it $(x_0,0)$ for instance, then the line $x=x_0$ makes perfect sense as a satisfactory definition.