No, obtaining a listing from ICC-SRCC or any other listing agency is generally voluntary (unless specifically required by a jurisdiction). However, a listing provides a manufacturer a proven way of demonstrating compliance with key codes and standards and can facilitate acceptance and approval in the field. It also provides a way of setting yourself apart from your competition.
Model plumbing codes do require third-party certification to the standards referenced by a code, but not to a code itself. For example, the 2021 International Plumbing Code (IPC) reads:
303.4 Third-party certification. Plumbing products and materials required by the code to be in compliance with a referenced standard shall be listed by a third-party certification agency as complying with the referenced standards. Products and materials shall be identified in accordance with Section 303.1.