Sounds somewhat like the Kombucha mushroom. Kombucha is a synergistic combination of various fungi and bacteria. It is generally grown on a sugar tea mixture where is converts the mixture in to a blend of beneficial acids. Acetic acid is the primary acid, which is vinegar. It helps to kill pathogens, control yeast growth, and aids in the absorption of nutrients. Hyaluronic acid, which is what holds our cells together. It is often referred to as the body's cement. Glucoronic acid, which is used by the body in the detoxification process and usinic acid, which is antibiotic and antifungal.
The kombucha creates a "baby" after a week or two that can be used along with the mother to create new batches of tea.
I have not heard of it being grown on milk, but I don't see why this could not be done since it is a sugar source for the fermentation.
I'll do a little research and see if I can come up with any more information.