Units don't match in the torsional spring energy!

Radians are a pure number, so they do not contribute to your dimension considerations.

The units of the torsion constant are $\mathrm{Nm}$ which are equivalent to Joules.

https://en.wikipedia.org/wiki/Radian#Definition

You are correct that, in terms of units, $k$ should be [E]/rad$^2$.

So why is it given as [E]/rad? Sloppiness/convenience.

The unit [E]/rad is equivalent both in dimension and in value to [E]/rad$^2$, so it probably makes life easier for engineers to pretend that $k$ is the same for both the torque $\tau=k\theta$ and the energy $U=\frac{1}{2}k\theta^2$. But they are not, which is something you would have to keep in mind when converting from radians to degrees.

Another motivation for this sloppiness is that it's presumably an attempt to emulate the standard harmonic spring $f=kx$ and $U=\frac{1}{2}kx^2$, for which the $k$ values are indeed equal.

An angle is just the ratio of the length of a circular arc to its radius, so the radian has units of length/length, which means it's a dimensionless quantity.