In many case it is an "extension" of the customer's specifications. They use ASME as the basis of material selection, design, fabrication, and inspection. However, they are not asking for it to be "code stamped" and the nameplate would not reference ASME.
You can't cherry pick the code if the vessel or the system has to meet code by contract or if it is required by the building code. If it is not required to meet code, the customer can pretty much dictate the terms.
Not all locations are required by law to meet ASME or National Board requirements. In those cases if their insurance carrier doesn't requires complience to ASME, there is nothing to force it upon the end user. A non-code vessel can have a nameplate with the design pressure and design temperature and any additional information requested by the customer as long as it doesn't say it is an "ASME Code Vessel".
Personally, I feel more comfortable when a third party is involved in the process. Although the AI represents the insurance company (most of the time), he is at least there to ensure the minimum requirements of the applicable ASME code section are met before he allows the vessel's nameplate to be stamped.
Best regards - Al