In the UK it would be pretty odd to see someone wear a hat indoors. You take it off when you come inside. On this side of the Atlantic, hats are usually seen as outdoor wear and you remove them – like you would sunglasses, a coat, or gloves – when you are indoors.
Americans, however, will often wear a baseball cap or beanie or whatever, indoors. I find it strange.
Mostly notice this when dealing with American tourists.