Have you ever seen https://www.law.cornell.edu/uscode/text/8/1449 8 USC 1449, that mentions both United States and United States of America in the same paragraph?

A person admitted to citizenship in conformity with the provisions of this subchapter shall be entitled upon such admission to receive from the Attorney General a certificate of naturalization, which shall contain substantially the following information: Number of application for naturalization; number of certificate of naturalization; date of naturalization; name, signature, place of residence, autographed photograph, and personal description of the naturalized person, including age, sex, marital status, and country of former nationality; location of the district office of the Service in which the application was filed and the title, authority, and location of the official or court administering the oath of allegiance; statement that the Attorney General, having found that the applicant had complied in all respects with all of the applicable provisions of the naturalization laws of the United States, and was entitled to be admitted a citizen of the United States of America, thereupon ordered that the applicant be admitted as a citizen of the United States of America; attestation of an immigration officer; and the seal of the Department of Justice.

If they both have to be mentioned, it presumes that both have separate identities.

Further inspection shows that the United States is NOT a member of the United Nations, however the United States of America is.