What does the potential of a charged conductor mean?

First, a very important point where you're wrong

Isn't electric potential defined for a point in space?

No, it's defined as the line integral of the electric field between two points $$V_{Arightarrow B} = -int_A^B vec{E}.text{d}vec{l}$$ The thing is that usually you omit to say which is the second point. For example, the usual expression of the electric potential for a point charge $$V(vec{r})=frac{1}{4pi epsilon_0} frac{Q}{r}hat{r}$$ express the difference of potential between $vec{r}$ (which represents $B$ in the above equation) and infinity.

Going to your question, as pointed in the comments, you can imagine a extended object as a bunch of points. In the case of a conductor, you know that all points have the same potential, so you only have to know the potential of one of those.

Finally, for circuits you usually say "potential of an element" when you're really talking about "difference of potential introduced by the element". For example, if your circuit have a battery of $12,V$, it means that if you take point $A$ of the integral at one side of the battery, and $B$ at the other, the difference of potential between $A$ and $B$ is $12,V$. Your expression can be interpreted in the same way.