The milk fat is easily washed away, so I would rule that out.
I think what you're seeing is the dried calcium of the tap water, especially if it seems powdery and can be rubbed off when dry. I think this is what user77907 means by scale. This is especially the case if your tap water is very hard.
You need not worry about these deposits: according to Wikipedia, the World Health Organization says that "there does not appear to be any convincing evidence that water hardness causes adverse health effects in humans."
I installed a water filter for drinking water (and for sterilization, coffee maker, etc.) in my kitchen, and all appliances as well as washed/sterilized items no longer have these deposits. If you're concerned about the purity of your tap water, I recommend this filter from APEC Water Systems; it's the best I could find, and many alternatives were much more expensive.