MySQL and PostgreSQL has JSON data types, is it the same as NoSQL database?
A relational DBMS processes data as relations, which is just a fancy way of saying tables consisting of rows and columns. Even if a column's type is defined as JSON, to the DBMS it is still a single column's value. There's an analogy with, say, a column defined as decimal. Externally we may say that column has an integer part, a fractional part, a number of decimal places and so on. To the DBMS, however, it is a single column's value.

The same holds even if code pulls values from within the JSON. These pulled values will be presented to the consumer as columns within a row. This row will be part of a relation (i.e. a table), and every row in that relation will have the same structure. Just because a query's result set is constructed on-the-fly at run-time it is no less a relation than those defined through a CREATE TABLE statement.

A NoSQL DBMS (and here I believe you mean the document DB variety, like MongoDB, rather than key-value or columnstores), however, thinks of data as JSON documents. It accepts well-formed JSON as input and returns JSON in response to queries. This is its model. It sees the world as documents, not tables.

In both models it is the interface between the client and the server that is the distinguishing feature. What each DBMS chooses to do internally is nobody's business but its.