PG doesn't stain, but from my experience with fog machines, the condensed vapor does become gooey when it deposits.
The nicotine itself should only yellow when exposed to UV, although I guess it could also oxidize. Regardless, the amount of nicotine in and exhalation is so minute in volume as to be a non-factor for practical intent.
PG/VG are humectants - they attract and bind water molecules. So the vast majority of what is being exhaled is water vapor, drawn from the body and from the environment - just like fog.
As observed, the exhalation from a cigarette that is the staining agent is depositing tar - and tar is the collection of toxic chemicals that are added to cigarettes. The filter is on a cigarette to catch the tar rather than the nicotine, and that's what stains brown or yellow-brown. Tar is very greasy which is what makes it so tough to get out even before the stain sets. So the appropriate degreaser for what you're cleaning should work best versus a standard cleaner, just as an fyi aside. [I never smoked inside so not facing that issue]