Why do we substract a background image and not divide it?
Division happens, indirectly, in cases where the image formation model is multiplicative. Concepts behind background varies on whether one only considers smooth fluctuations, or one includes noise. For the later, there is a lot of literature on multiplicative noise.


  important example(s]: speckle noise [...] proper shadows due to
  undulations on the surface of the imaged objects, shadows cast by
  complex objects like foliage and Venetian blinds, dark spots caused by
  dust in the lens or image sensor, and variations in the gain of
  individual elements of the image sensor array.


For a combined case, the principle of homomorphic filtering considers a nonlinear transformation on the image $f$ to cope with two main components (one slow, illumination $i$ and one fast,  reflection $r$). In the most basic case, $f=i\times r$ is passed through a logarithm before some linear filtering (and an inverse exponential on the results).This allows to reduce the noise without compressing the dynamic range too much. 



It is suitable for instance to shading effects on surfaces, where one end is darker than the other, for instance, from Homomorphic filtering – part 2 (Mathworks):



In the mathematical morphology community, you can also find the LIP concept, for Logarithm Image Processing.


Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization
Logarithmic Image Processing: Additive Contrast, Multiplicative Contrast, and Associated Metrics
Image restoration with multiplicative noise
Side work: [Multiplicative Calculus in Biomedical Image Analysis](https://doi.org/10.1007/s10851-011-0275-1