Dimensions and units are separate things, though. For example, 1 minute and 1 second are different units of the same time dimension. Similarly, 1 rad and 1 degree are both dimensionless, but they are both different units.

It is only equal to 1 by convention. If we instead considered the ratio of the diameter to the arc-length then rad would be 1/2.

Perhaps in the physics sense, but in computer science we do have the notion of types which does allow us to model the difference between an angle and other numerics.