How can I define multiple identifier classes and styles in listings?

You're halfway there. The listings manual (p.31 in v1.5b) tells you that multiple classes of identifiers and associated styles can be defined by using

\emph=[<number>]{<identifier list>}
\emphstyle=[<number>]{<identifier style>}

where <number> is some integer of your choice that gets associated with the class of identifiers in question.

\documentclass{article}

\usepackage{xcolor}
\usepackage{listings}

\lstset%
{%
    emph=[1]%
    {%
        DELETE,
        GET,
        POST,
        PUT,
    },
    emphstyle=[1]{\color{green}},
    %
    emph=[2]% Variable Types
    {% 
        blob,
        datetime,
        integer,
        string,
    },
  emphstyle=[2]{\color{blue}},
}

\begin{document} 
...
\begin{lstlisting}
    {
        "id": {integer},
        "login": {string},
        "password": {string},
        "name": {string},
        "picture": {
            File Resource
        }
    }    
    ...       
    PUT /files/{file_id}
\end{lstlisting}
\end{document}

A supplement to @Jubobs's answer as a reminder for users of listings v1.6 (maybe also for versions later on, I haven't checked yet).

In listings v1.6, the valid syntax for specifying group in emph and keywords has been changed into

emph={[2]...}, emphstyle={[2]...}
keywords={[1]...}, keywordstyle={[1]...}

These can be found in the listings manual (v1.6, p22).