Any sort of registration is only to make it easier to “prove” that the person is a clergy, which makes it easier to prove that the marriage took place. Some say that in practice, all they need is a letterhead.
American law has a rule called the “mock priest” rules going back to the middle ages that means that even if the wedding is conducted by a “fake” clergyman, it doesn’t affect validity of the marriage. But as with other unusual ways to get married (e.g. filing a marriage contract) there are reasons no one relies on it.
The easiest way to get married is by getting a license, and using a clergy whom the local government will readily recognize as such. While any other sort of wedding is valid under American law, it involves much extra effort and expense in order to get the wedding recognized by the government.