The background to the note in the instructions is that milk residue can deposit in the small gaps between the silicon seal, glass container and plastic case, which is much more difficult to remove than for example coffee residues.
The reason for this is the varying consistency of drinks. Coffee is a water-based drink and can simply be removed with water.  Milk is a fat-based drink, therefore it is thicker and not as easy to remove and clean with water.
Milk residue is ideal breeding ground for bacteria and pathogens. To prevent this health risk, we recommend that you do not store milk products in the flasks. This also applies to other branded plastic flasks.