Angles are a peculiar case, and don't really fit the pattern of other physical quantities. But the SI brochure isn't giving a complete picture of what's really going on here. It's quite complicated.
Physical quantities are characterised by their symmetries. The physical quantities exist in their own space, and there are symmetry operations that map them on to one another while preserving, in some sense, the physics. We can build a coordinate system for these physical quantities by picking a 'standard' example, and then specifying any other instance of the physical quantity by listing a set of symmetry transformations needed to get to it from the 'standard' value. The symmetry transformations can be parameterised by various sorts of 'numbers' (that might or might not be the <PERSON>).
This is called a 'homogeneous space' by the mathematicians. If there is a unique transformation to get to any other instance (so the coordinates are unambiguous and well-defined), then it is called a 'principal homogeneous space' or 'torsor'.
Many physical quantities have a scaling symmetry. So we can specify any other quantity by scaling the standard unit by a scale factor. This turns the symmetry group action on some abstract space of physical quantities into a vector space over the Reals, which is much easier to handle.
Angles don't have a scaling symmetry, they have a rotational symmetry, which is periodic. A rotation of $2\pi$ radians is the same as the identity, so there is a natural 'scale' already defined for them. The symmetry is already broken.
However, we can still find a mapping between the angles and the Reals, by wrapping the Real line around the unit circle infinitely often. And then we can apply scaling transformations to the Reals anyway, despite the fact that it breaks the symmetry. That's how we get 'degrees' and 'radians' as alternative angular 'units'.
This mapping uses the Lie algebra for the rotation group. It works a bit like a generalised logarithm. 855
The rotation $\mathrm{exp}(i\theta)$ has logarithm $i\theta$ where we can treat $\theta$ as a Real number and $i$ acts like a 'unit' for rotation. In fact we can generalise this to more than two dimensions by setting $i$ to be a bivector, a quantity that specifies an oriented plane element with magnitude, and has the property that its square is -1. This is actually constructed as a ratio of two vectors pointing in different directions, and rather than just setting a size it also specifies the orientation of the plane in which the angles lie.
As a historical note, this was how <PERSON> invented the Quaternions - as the ratio of a pair of vectors in different directions. He had hoped to be able to use them to create an algebra for 3D vectors in the same way Complex numbers worked for 2D vectors, but he was never able to get it to work properly because bivectors don't work the same way as vectors. (Complex numbers aren't vectors, either, for the same reason.) The two can be unified in Clifford algebras, but that's another story.
So, there is an analogy here between scaling transformations and rotations, in that one can be implemented by multiplying by $\mathrm{exp}(\lambda)$ and the other is implemented by multiplying by $\mathrm{exp}(i\theta)$. Multiplication is turned into addition, and a change of unit becomes an additive offset. The choice of unit in these coordinates is effectively a choice of origin.
Thus we can see in several ways that the algebra of linear units is not the same as the algebra of angles. We can't use the same "scale a standard unit" trick to represent them. However, this does not mean that angles are Real numbers, either. We can represent them as unit Complex numbers in 2D, or as unit Quaternions in 3D, but using the multiplication operation to combine them instead of addition. (Or we can take the logarithm to get an additive operation, at the cost of making the mapping periodic and thus ambiguous.) But whatever we do, we're not getting the same sort of object as the dimensionless ratio of two scalar lengths - a Real number.
This isn't quite the end of the story. If we consider an angular velocity, we find that the original $0=2\pi$ identification no longer applies. $2\pi$ radians per second is physically different from 0 radians per second. We have to work now in the Lie algebra, which can now be thought of as the tangent space to the rotation group at the identity. Angular velocity is thus properly a bivector, although in 3D we can use the dual to it and treat it as a vector. 364