The SQL Standard, in the part called SQL/JSON, describes syntax, yes. Some DBMSen have it implemented already (for quite some time). PostgreSQL has the functionality but not yet the required syntax.
(In fact the implementation of that syntax for PostgreSQL is waiting to be committed, so I suppose it will be in version 15, next year. It works quite well, and I was able to use the ORACLE and DB2 documentation for SQL/JSON when I was trying it out and looking for examples of that standard-compliant syntax)
There is a link to a SQL/JSON pdf (zipped) in the Bibliography part of the PostgreSQL docs:
update: fixed typo's!