I know that the US began as 13 colonies of the UK and many aspects of the US were inspired by the UK, so I'm wondering if the name is one of these aspects.
No. At the time of the American Revolution, there was no "United Kingdom." The "Kingdom of Great Britain" and the "Kingdom of Ireland" were independent Kingdoms ruled by the same monarch. The entity that we now know as the United Kingdom did not exist until the two kingdoms merged into one "United Kingdom of Great Britain and Ireland" with the Acts of Union of 1800.