The problem is that Source Sans Pro Italic does not contain most Greek characters. It does have π, so your example will work with \pi
to illustrate this. You can also confirm this by typing Greek letters into the sample section of the web page you link to.
You could use upright characters instead of italic characters for Greek in maths:
\setmathfont(Greek)[Uppercase=Regular,Lowercase=Regular]{Source Sans Pro}
Or you could use a different but similar font, e.g., Noto Sans:
\setmathfont(Greek){Noto Sans}
MWE with Source Sans Pro Upright Greek
\documentclass[a4paper,10pt]{article}
\usepackage{mathspec}
\defaultfontfeatures{Mapping=tex-text}
\setmathrm{Source Sans Pro}
\setmathfont(Digits,Latin){Source Sans Pro}
\setmathfont(Greek)[Uppercase=Regular,Lowercase=Regular]{Source Sans Pro}
\setmainfont[
Scale = 1.0,
BoldFont = *-Semibold
]{Source Sans Pro}
\begin{document}
aAbBcCdDeEfFgGhHiIjJkKlLmMnNoOpPqQrRsStTuUvVwWxXyYzZ
123456789
\textbf{Bold}
\textit{Italic}
\textbf{\textit{Bolditalic}}
$math$
$\gamma \rho \epsilon \epsilon \kappa$
\end{document}

MWE with Noto Sans
\documentclass[a4paper,10pt]{article}
\usepackage{mathspec}
\defaultfontfeatures{Mapping=tex-text}
\setmathrm{Source Sans Pro}
\setmathfont(Digits,Latin){Source Sans Pro}
\setmathfont(Greek){Noto Sans}
\setmainfont[
Scale = 1.0,
BoldFont = *-Semibold
]{Source Sans Pro}
\begin{document}
aAbBcCdDeEfFgGhHiIjJkKlLmMnNoOpPqQrRsStTuUvVwWxXyYzZ
123456789
\textbf{Bold}
\textit{Italic}
\textbf{\textit{Bolditalic}}
$math$
$\gamma \rho \epsilon \epsilon \kappa$
\end{document}
