The way I would approach this is to consider blackbody radiation, where a spherical blackbody radiator is isotropic, e.g. it emits radiation equally in all directions. If we were to model the noise introduced by such a blackbody, it would be additive Gaussian white noise. In this example, it is called white noise because it has a flat power spectrum density, and it is Gaussian because the noise has Gaussian amplitude distribution. It is additive because it can be added linearly to the desired signal.
In this case isotropy is a characteristic of the emitter. If I would interpret isotropy in terms of noise, I would understand it as being the same in all directions with respect to the receiver, e.g. regardless of direction, the received signal would have identical noise functions that were added to the signal. If the noise had some constant associated with it, as long is the constant was additive and the same in all directions with respect to the receiver, I would consider that isotropic as well.