Is the term hanger or coat hanger used more often in colloquial speech in American English? I'm afraid Google Ngram is of no help here, as the first word has several meanings, plus we are talking about colloquial speech here.
Coat hanger, clothes hanger, and just plain hanger are all used. Largely the choice depends on the context and the hanger style. The things in the entry way closet are coat hangers, the ones in the bedroom closets are just hangers, and the ones on the racks at J C Penney's are clothes hangers.
Normally when you say "coat hanger" you are referring to a heavier-duty hanger intended for coats and other heavy garments. Likewise, a "clothes hanger" is more likely to be a bit on the heavy-duty side. A plain old "hanger" is generally going to be the flimsy wire thing.
I'd have put this in comments, but I don't have enough points yet.
I'm from Michigan, and we generally just say "hanger" for whatever type of hanger. The only time someone would say "coat hanger" would be to indicate the hardier type of hanger that you could use to hang your coat. For example, I might tell someone there are hangers in the closet they can use. When they open the closet to hang up their coat and see a bunch of regular hangers, they might say, "Do you have a coat hanger I can use?" If there had been a coat hanger or coat hangers in the closet, though, it still would not have been weird for me to refer to them as simply "hangers."