Before America declared its independence in the 18th century, and before George Washington became the country's first President, did the country have a monarchy? And if so did it have the same monachy as the United Kingdom - i.e. was it part of the then Commonwealth?
At the time your talking about, America was part of the British Empire, and their King was the same as ours, George 111, if memory serves me right. I haven't googled this, so if i'm wrong, i'm sure somone will put a correction, but I am pretty sure. As for the Commonwealth, that wasn't around then, it was a progression of the Empire, whereas countries couldn't voluntarily leave the Empire, they could, and can, leave the Commonwealth.
Like every other English colony of the time, a small part of what was to become, eventually, the United States of America recognized and were governed by the English Crown... mad King George III was king of Great Britain and as such wielded the power of the Crown. Keep in mind that other European monarchies held claim to vast areas of the North American continent at the time as well... France and Spain come immediately to mind...
...indeed the French, with their historical enmity towards the British, were instrumental to the outcome of the War of Independence.
How would history have changed if George Washington had achieved his goal to secure a commission as a regular officer in the British Army instead of being passed over for promotion and remaining a colonial militia officer?