$e^{i\theta} = 1$ for all $\theta$? What is the conceptual error in the following derivation?
$$
e^{i\theta} = e^{i\frac{2\pi}{2\pi}\theta} = (e^{i2\pi})^{\frac{\theta}{2\pi}} = 1^{\frac{\theta}{2\pi}} = 1
$$
It is clear to me that the second move is illegal, I just don't know why.
$a^{x y} \not = (a^x)^y$ in general.

For example, $[(-1)^2]^{1/2} = 1$ but $(-1)^1 = -1$.

You may wish to look up "branch cuts" to understand conceptually why this is true.
$\theta$ is the angle from the real axis so what you did that while converting $e^{i\theta}$ to $e^{i\theta \frac{2\pi}{2\pi}}$ you change the complex number because if $\theta \ne 2\pi$ then $e^{i\theta \frac{2\pi}{2\pi}} \ne e^{i2\pi \left(\frac{\theta}{2\pi}\right)}$ because the argument of the $e^{i\theta }$ is $\theta$ and the argument of $e^{i2\pi}$ is $2\pi$