People in the USA will know what you are talking about, but no - "like anything" is not used often. 

Like anything? I have never heard that phrase. 

 "It's raining like cat's and dog's." is common.

I've seen that a lot of things are used to compare. He studied like a horse, for example. The "horse" is used for emphasize a vast content, probably due his big aspect. When I want to compare something with a word and it doesn't pass through my mind, I usually say: "It's raining like I don't know what!".