Would 2 C₂ be 2 moles of diatomic carbon, or 4 moles of carbon?
You can have 1 mole of $\ce{C_2}$ which contains 2 mole of $\ce{C}$ just as you can have for example 1 mole of $\ce{C_2H_5OH}$ which contains 2 mole of $\ce{C}$, 6 mole of $\ce{H}$ and 1 mole of $\ce{O}$.

The point is that you always have to mention whether you are indicating moles of molecule or moles of atoms. That is why a statement like: "I have 1 mole of oxygen" is ambiguous, because it could be referring to oxygen gas (i.e. $\ce{O_2}$) or to atomic oxygen (i.e. $\ce{O}$).