Wikipedia's double-and-add point multiplication algorithm for elliptical curves
P is an x, y coordinate and so it seems reasonable to assume that 0 is a 0, 0 coordinate


Actually, that's an incorrect assumption; by 0, they really mean the point at infinity (aka the neutral element aka the identity element).  That is, it is a point that doesn't actually correspond to a solution to the cubic equation, but instead is something that needs to be thrown in to make the curve a mathematical group.

As for how it is represented, well, that rather depends on the implementation; there might be implementations that represent it as (0, 0), but (as you point out), that might not work if $x=y=0$ is a valid solution to the equation.