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According to this answer, the units for angular velocity squared are $mathrm{rad}/s^2$. The units for angular acceleration are also $mathrm{rad}/s^2$. Why is this the case?

They have the same units of $mathrm s^{-2}$ only if you don't use $mathrm{rad}$ unit for bookkeeping (which you can indeed avoid because radians are technically dimensionless, similarly to turns and other auxiliary units).
But if you do try to distinguish angles from dimensionless numbers, then they are not the same: the unit of angular acceleration is $mathrm{rad}/mathrm s^2$, and that of square of angular velocity is $mathrm{rad}^2/mathrm s^2$.

If radian is dimensionless, why do we not "simplify" $mathrm{rad}^2$ to $mathrm{rad}$? For the same reason as why we introduced $mathrm{rad}$ in the first place: it's not required to use this symbol, but it does help us remember that we have an angle somewhere. Similarly, if we square it, we must use $mathrm{rad}^2$ because now we have a square of that angle. But as the unit is dimensionless, it's technically not necessary. It's simply for convenience. We could invent a bunch of other dimensionless units to aid us in bookkeeping, but once we've done it, we must keep their correct powers, otherwise these units are simply useless.

By definition, angular velocity is defined as the rate of change of angle with respect to time, leading to the equation $omega = Delta theta / Delta t$. From dimensional analysis, this yields units of radians/s. Also by definition, angular acceleration is defined as the rate of change of angular velocity with respect to time, leading to the equation $alpha = Delta omega / Delta t$. From dimensional analysis, $Delta omega$ has units of rad/s, so the units of $alpha$ are $rad/s * 1/s$, leading to final units of $rad/s^2$.

Note that directly comparing units of $omega ^2$ to units of $alpha$ with no physical basis for doing so, doesn't make sense from a physics standpoint.

They have the same dimensions because radians are adimensional. The only physical dimension is time so when you square it you get a $time^{-2}$ and when you derive you get the same.

Radians are dimensionless. You can safely set $text{rad}to 1$. Using $[x]$ to mean "the units of $x$, we find:

The units of angular velocity $[omega] = left[frac{dtheta}{dt}right]= frac{1}{T}$ where $T$ is time. Then $[omega^2] = frac{1}{T^2}$.

The units of angular acceleration $[alpha] = left[frac{d^2theta}{dt^2}right] = frac{1}{T^2}$.

Your question is a bit like why does my decimal representation of 1/3 never terminate? It's because we have chosen base 10 and if we choose a different base (e.g. 9) we can make 1/3 terminate (e.g. 0.3).

Units are just comparing a quantity to a reference quantity chosen by convention. I could use seconds as a distance measurement, where it was understood that the reference distance was how far light travels in a second. In that case distance and time have the same units and distance becomes dimensionless akin to refractive index.

We choose units to keep track of what references we have used, hence the use of rad even when it is dimensionless, and also why we use metres instead of seconds for distance. Units are much more conventional than we often take them to be. Look at the rash of unit systems used in electro-magnetism.