Are angles ever multiplied?

Perhaps this is a bit too obvious to be a useful answer, but I will post it anyway:

Angles are only defined up to congruence $\!\!\mod2\pi$. Addition preserves this symmetry: $$ (\vartheta+2\pi k) + (\varphi+2\pi j) = \varphi + \vartheta + 2\pi(k+j) =: \xi + 2\pi l, \qquad k,j,l\in\mathbb{Z} $$ while multiplication doesn't. Therefore, the multiplication of two angles cannot be well-defined, unless you come up with some additional constraint on the range of the angles (in physics, you would call it a gauge). But this would not represent the mathematical/physical meaning of angles.

For what it's worth, there is such a thing as a "square degree" and a "square radian", better known as a steradian. As GEdgar notes, these come up in spherical trig.

Maybe look in spherical trig for more. Area of a spherical triangle, and such things.