If I draw any kind of shape on a piece of paper (possible discontinuous), would it be theoretically possible to represent the shape in a mathematical function?
You have to be careful here. By definition, a function has one possible output for any given input. So if you want your function defined as some $y = f(x)$, then no not every shape can be written as a function. Any shape that has two points directly above each other (relative to the x-axis) cannot be written as a function, even a piecewise one.
However, if you are defining "a function" as any parametric curve $(x,y) = (f(s),g(s))$, then yes this can be done for any shape that can be drawn without lifting your pencil off of the paper. Simply define your parameter $s$ to run along the same path your pencil moves and your $f$ and $g$ are therefore well-defined too.
As a function? No because no function can have more than one output value. This means that if your draw a line parallel to the y-axis, the line should cut the function at one and only one point.
But you will always find a relation that gives the image. And there's always an equation for every picture if you don't constrain your choice of functions, you can use, too much.
For example, there's no function whose graph is a circle but the equation $$x^2+y^2=1$$ gives a circle of radius 1.