You can add >>> with morekeywords but you also have to change > to be a letter with the alsoletter option for it to work:
\documentclass{article}
\usepackage{listings}
\usepackage{xcolor}
\lstset{
language = Python ,
columns = flexible ,
escapeinside = {<@}{@>} ,
frame = lines ,
alsoletter = > ,
morekeywords = {>>>}
}
\begin{document}
\begin{lstlisting}[caption=Python Console, label=amb, numbers=none]
>>> print(Hello World)
\end{lstlisting}
\end{document}

If you want it blue but not all keywords blue you can add it to a different class of keywords and set a style for said class by using the optional argument of the corresponding options:
morekeywords = [2]{>>>} ,
keywordstyle = [2]\color{blue}\bfseries

As soon as you add \usepackage[T1]{fontenc} (which in general is a good idea) to the example you won't be happy anymore, though:

>> is turned into » (as a ligature) then. In that case personally I'd use teletype as font for the code (I'd do that in any case, but that's just me) with basicstyle = \ttfamily:

Since now the keywords no longer are bold we need to use a teletype font that has a bold series, eg with \usepackage{lmodern}:

If you don't want teletype code but still want to use T1 encoding and if you don't need >> to be treated as a ligature anywhere else in your document you could also add the microtype package and disable the ligature
\usepackage{microtype}
\DisableLigatures[>]{encoding = T1}
(Using the microtype package will do your document good, anyway…)
without teletype:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{microtype}
\DisableLigatures[>]{encoding = T1}
\lstset{
language = Python ,
columns = flexible ,
escapeinside = {<@}{@>} ,
frame = lines ,
alsoletter = > ,
morekeywords = [2]{>>>} ,
keywordstyle = [2]\color{blue}\bfseries
}
\begin{document}
\begin{lstlisting}[caption=Python Console, label=amb, numbers=none]
>>> print(Hello World)
\end{lstlisting}
\end{document}

with teletype:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{microtype}
\DisableLigatures[>]{encoding = T1}
\lstset{
basicstyle = \ttfamily ,
language = Python ,
columns = flexible ,
escapeinside = {<@}{@>} ,
frame = lines ,
alsoletter = > ,
morekeywords = [2]{>>>} ,
keywordstyle = [2]\color{blue}\bfseries
}
\begin{document}
\begin{lstlisting}[caption=Python Console, label=amb, numbers=none]
>>> print(Hello World)
\end{lstlisting}
\end{document}

>symbol does not occur in the default text font (computer modern), only in the symbol font. you can get the symbol with\texttt{>>>}without entering math mode, if you're happy with the "typewriter" style. – barbara beeton Aug 26 '16 at 14:17