Pounding one's fist on something hard is a natural expression of anger. This expression is often used figuratively, although I imagine that some people (Hulk Hogan?) are capable of doing it literally.
I think it is literally what happened. He punched the door so hard that he broke it, and probably broke some bones in his hand too. [EDIT] Interior doors in American homes are hollow, made of very thin wood. They are not very strong.
I have not seen the expression used as an idiom. I have seen text like "I was so angry that I could have put my fist through the door."
If an American English speaker says he "put his fist thru a door" in reaction to something, he may or may not be expressing an actual event. I would expect it to be an exaggeration for humorous effect. However, I have known of someone who kicked a hole in a door in anger.