Why are the units of angular acceleration the same as that of angular velocity squared?

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.