The traditional, 1000's of years old soaps (sodium or potassium salts of fatty acids, used at least as early as ancient Egypt) are perfectly eddible not only for bacteria, but for mammals as well. Ever had your dog eat your soap?
Well, they may be too much alkaline for anyone's taste (or homeostatic equilibrium), but this can be compensated or in some cases lived with. The soaps used in the household or for hygiene purposes are never alkaline enough to burn your skin, so they are safe-ish for microorganisms as well.
Given enough water, these substance pretty much hydrolyse by themselves. The thin, barely visible layer the soap leaves over the water surface is pretty much the pure acid (the sodium ions are happy to dissolve to the bulk of water).
Other detergents (sulfonates, phosphates, etc...) have different compositions, but their common characteristic is the large non-polar part of their molecule that is still separable as a fatty acid. Om-nom-nom. The polar part of the molecule may be itself useful for microorganisms in one way or another (e.g. phosphates are of particular interest because they fertilize the open water bodies and algae grows more than it is acceptable).
Major detergent markets (EU, USA) even require biodegradability of the household detergents. The economy of scale means that you will get biodegradable detergents even outside of these markets. Nothing stops bacteria from starting biodegradation right in your kitchen sink (the osmotic pressure may slow them down, but only that much).
On the other hand, these microorganisms that can eat detergents cannot be really blamed for being pathogen for humans. If you are really lazy, leaving the dishes soaked can even help you do them later. The only real problems are the messy look, the smell and your life choices.
What happens to the microorganism's precious lipids? Given enough concentration of the detergent, the lipid layer will decompose. This is (one of the mechanisms of) how your skin dries when you do too much dishes. On the other hand, the outer cellular borders are not only lipids. There are proteins, glycans and other things that together with lipids make the cell not so much vulnerable.
The main factor one may expect to act on microorganisms subjected to common detergents is the osmotic pressure. The cells simply lose water. This slows down all the processes inside.