When any of the 3 output variables depends non-linearly of all 3 input variables that is not true.
Even on something as simple as a sphere, you can have a spherical triangle where one side is longer than the sum of the 2 other sides (e.g. when one side is on the equator and the 2 other sides go to one pole and the arc on the equator is greater than a half circle).
And you can map a (partial) sphere to a (partial) plane (mapping e.g. an equator segment and some meridian segments to straight lines, in order to map some plane triangles to some spherical triangles) and then use the distance in the plane as a difference function for pairs of points on the sphere.
Suppose we have a 'difference function' D on the space of colours, a simple metric d on the target space, and some nonlinear function f from the space of colours to the target space such that D(x,y) = d(f(x),f(y)) for all x and y.
Then the triangle inequality for D would be D(x,y) + D(y,z) ≥ D(x,z). By the equation above this is equivalent to d(f(x),f(y)) + d(f(y),f(z)) ≥ d(f(x),f(z)), which is true by the triangle inequality in the target space at the points f(x), f(y) and f(z). If the target space uses a simple metric like the Euclidean or Manhattan distance then the triangle inequality is always going to hold in the target space and hence in colour space.
Note that the above argument does not need linearity of f.
The sphere example doesn't hold up because if two points are further apart than half way around a great circle then the distance between them is actually the length of the path going the other way.
I think that you are right, and my answer had been too hasty.
What can be obtained experimentally is only the difference function D(x,y).
If it cannot be decomposed into a distance function and a space transformation function, i.e. as D(x,y) = d(f(x),f(y)), then that means that the quest for an uniform color space, which has seen a very large number of attempts during a century (the quoted Schroedinger paper is from 1920), will never produce a completely satisfactory result and that the best results for color interpolation or for color matching within tolerances can be obtained only by using a linear RGB or XYZ space together with a complicated non-linear color difference function.
Even on something as simple as a sphere, you can have a spherical triangle where one side is longer than the sum of the 2 other sides (e.g. when one side is on the equator and the 2 other sides go to one pole and the arc on the equator is greater than a half circle).
And you can map a (partial) sphere to a (partial) plane (mapping e.g. an equator segment and some meridian segments to straight lines, in order to map some plane triangles to some spherical triangles) and then use the distance in the plane as a difference function for pairs of points on the sphere.