With so much shared culture and language it's easy to feel as if the UK and the USA are much closer and more aligned than they really are. The culture difference is absolutely vast, especially when it comes to freedom and rights. We do not see guns on a daily basis, and when we do they're in the hands of a particular type of police officer. Standing in line in a store in the USA behind someone with a pistol or assault rifle is really strange.
Conversely, while most of us are numb to the fact that we live under a monarchy, in the USA it's a quaint peculiarity that's fun to observe from the outside, or to use to make a point about independence.