Why does tea dissolve better when put in water rather than milk?
Milk is an emulsion containing small bubbles of oil/fat. Whenever you boil milk or do something drastic, this fat tends to separate. You can do this nicely, and get cream or butter, preserving the bubbles, only part of the water content is separated, or can do more drastically, when these bubbles collapse to a big oily blob.

When you put your filter in the milk, most probably some of the oil/ fat bubbles hit the surface and cover it with an oily film. This film prevents the easy dissolution of caffeine, etc from the tea leaves in water.