Coats of arms are individual. The idea that everyone with a certain family name is entitled to the "family" coat of arms is something invented by people who are selling fancy plaques and stuff with the "family" arms to whoever wants to buy them. A given coat of arms belongs to a single person. There are rules for how the arms are passed down generations, and how the arms are modified for use by the first son, second son, daughters, etc.
It is also incorrect that only royals had coats of arms. Many people had/have then, everyone down to the rank of knight and Baronet, and a lot of people below that.
English Coats of arms were, and still are, given by the College of Arms. There is an American College of Arms, but they just register arms claimed by people, they do not oficially check or award arms.
But, since we fought a whole war about breaking with the idea of the King telling us what to do, we don't have to pay attention to the English rules! :grin:
Also, we can all bear arms, the second amendment says so! :haha: