I think Oracle does. My impression is that PostgreSQL sort-of "invented" JSON support in the db, then Oracle (and probably others) added JSON support with a different syntax, and then Oracle got their version defined into the SQL standard. I'm half-guessing here, but it's based on what I've gleaned from Markus Winand's [0] excellent compatibility tables in his slides [1].
SQLite (the most popular RDBMS today) doesn't, unless you count extensions. And of course, none of them use the same syntax or functions or data types.
If I recall correctly, Oracle's support for JSON is not at the same level as PostgreSQL. By that I mean seamlessly store JSON as a data type like any other. In Oracle JSON is stored as VARCHAR/CLOB and then you implement tests to validate whereas in PostgreSQL and MSSQL is its own data type.
(Last I checked atleast) JSON isn't as well supported on MySQl or MSSQL compared to PG.
In PG, a JSON column is so well integrated that you can do all sorts of crazy stuff (indices over JSON queries is my favorite). You could build an entire RDBMS on top of PG's JSON column.
Sure, but the non-standard enhancements like JSON support are part of what sets Postgres apart from the competition IMO.