XeLaTeX, fontspec and printing the name of the current font

You can access the internal and the external names of the current font respectively with \the\font and \fontname\font. The former must be stringified in order to print it.

Further massaging of the external font name can be added.

\documentclass{article}

\usepackage{fontspec}

\setmainfont{Libertinus Serif}
\setsansfont{TeX Gyre Adventor}[Scale=MatchUppercase]
\newfontfamily{\junicode}{Junicode}

\DeclareTextFontCommand{\textttup}{\normalfont\ttfamily}

\newcommand{\printinternalcurrentfont}{%
  \expandafter\textttup\expandafter{\expandafter\string\the\font}%
}
\newcommand{\printexternalcurrentfont}{%
  \expandafter\textttup\expandafter{\fontname\font}%
}

\begin{document}

\printinternalcurrentfont

\printexternalcurrentfont

\bigskip

{\itshape\printinternalcurrentfont\par\printexternalcurrentfont}

\bigskip

\sffamily

\printinternalcurrentfont

\printexternalcurrentfont

\bigskip

\junicode

\printinternalcurrentfont

\printexternalcurrentfont

\end{document}

Output with XeLaTeX

Output with LuaLaTeX

If you're not interested in the finer details, but just in the main font name part, change the definition of \printexternalcurrentfont into

\makeatletter
\newcommand{\printexternalcurrentfont}{%
  \expandafter\format@externalcurrentfont\fontname\font:\@nil
}
\def\format@externalcurrentfont#1:#2\@nil{%
  \textttup{\@ifnextchar"{\@gobble}{}#1}%
}
\makeatother

The output would become