Why is the "drinking milk" trope so prevalent in anime?
There's probably no one answer for why they drink milk in general, but as far as the bathing, it's considered a traditional drink in Japanese public bath houses after a soak:


  There is usually a refreshment cooler here where customers can self-serve and pay the attendant. Milk drinks are traditional favorites and sometimes there is ice cream.


Most places don't know how the tradition started but it's been around for a long time and some people say that milk drinks (not just plain milk but also milk+coffee) taste particularly good after a hot bath.