The Auxiliary Manual authorizes members to wear a name tag with the first name included in lieu of the standard name tag (last name only), but this only applies to name tags of Bakelite or similar material such as those worn on the Tropical Blue uniform, not to cloth name tapes.
Here is the pertinent excerpt from the Auxiliary Manual:
“The Coast Guard Uniform Regulations, COMDTINST M1020.6 (series) authorize active duty and Reserve personnel to wear name tags that display first and last names in lieu of the standard name tag (last name only). An individual‘s first and last name and the U.S. Coast Guard legend letter sizes must be the same as for the standard name tag. This applies only to name tags of Bakelite or similar material, not to cloth name tapes. Commands have the authority to require the standard name tag when a higher level of uniformity is necessary. Auxiliarists are similarly authorized to wear name tags with first and last names displayed in accordance with these guidelines. A displayed first name must be the Auxiliarist‘s full common first name or conventional shortened derivative (e.g., Mike instead of Michael). Nicknames or parenthesized names are not authorized (e.g., Doc, Mac, “Lefty”). Auxiliarists must also have a standard name tag for when a higher level of uniformity is necessary.”
Auxiliary interpreters may wear a pin-on Auxiliary interpreter name tag with the Service Dress Blue and Tropical Blue uniforms during authorized interpreter missions. It shall display the full first and last name in the first row and the word “INTERPRETER” in the second row in place of the traditional words “U.S. COAST GUARD AUXILIARY”.
Source: COMDTINST M16790.1G Auxiliary Manual, Ch.10.F.3.