The requirement for registration flows from the legal jurisdiction where the vessel will eventually be in-service.
For example, in the Canadian province of Alberta, there is legislation called the Safety Codes Act, that requires compliance with CSA B51, which in turn requires registration of pressure vessels with the local jurisdiction, in this case, administered by the Alberta Boiler Safety Association (ABSA). So, the requirement for registration comes from the "LAW". In jurisdictions that do not require adhering to the ASME Standard (note that it only becomes a Code when brought into force by a law), there is no legal requirement to register the pressure vessel. However, the company's insurance carrier may require it, and it could be interpreted that the OSHA requirements (in the US, at least), require adherence to the ASME Standard.