Why isn’t math bold when set in bold context, like \textbf{Text $f_i$}?

The f isn’t bold because it’s set in math mode and in math f  and f  can be different things. E.g. f  is a vector and f  the length of the vector.

LaTeX has its own treatment for math fonts which can be affected by the user with commands like \mathbf. But the math fonts don’t depend on the surrounding text font. You may have a look at Make mathfont respect the surrounding family, where you can find settings to get your desired behaviour. An image taken frome there to illustrate the result:

Please post complete documents not just fragments (so this isn't tested) but you want

 \begin{document}
 \textbf{\boldmath\textit{Les caractéristiques $f_i$}}
 \end{document}

Just to mention all possibilities, there is also the bm package with its \bm command:

\documentclass[12pt,a4paper]{book}
\usepackage[utf8]{inputenc}
\usepackage{bm}
\begin{document}
\textbf{\textit{Les caractéristiques $\bm{f_i}$}}
\end{document}