If $x=a$ is not a function, nor an equation—as in its graph is not the set of all points which satisfy that equation—, then what is it? In precalculus, I was introduced to conic sections and their equations. I learned how parabolas, for example, aren't always formed from quadratic functions and how they're oftentimes better described with the use of equations. A parabola is then better defined as the set of all points which satisfy either of the two following equations:$$(y-k)^2 = 4p(x-h)$$ $$or$$ $$(x-h)^2 = 4p(y-k)$$ That being the case, what exactly is $x=a$? It's not a function because it's not 'one-to-one' and it's not the set of all points which satisfy that equation because if that were the case then its graph would be a single coordinate-point, $(x, a)$. If it's not a function, nor an equation, then what is it? Why is $x=a$ graphed as a vertical line, and likewise, why is $y=b$ graphed as a horizontal line?

$x = a$ is certainly an equation, because it's of the form (left side) = (right side)
where the left and right sides are expressions.  Assuming $a$ represents a real constant, its graph is the set of all $(x,y)$ where $x = a$, and that is indeed a vertical straight line.
