One of the things necessary to describe things like mean and standard deviation is that our random variable live in a domain like the real numbers.
Angle does not live in a domain like the real numbers. It has the cyclic property that 0 = 2π. In particular, division is not a well-defined operation on angles... but division is an essential part of computing means and standard deviations.
As an example, what should the average of north and south be? We could say that north is π/2 radians and south is 3π/2 radians, and say that the average should be π (west)... however if we said south is -π/2 radians, then the average would be 0 (east).
Standard deviation is even less well-defined; not only does it rely on knowing what the mean is, it uses both division and square roots; square roots are very poorly defined for this domain!