It's easy enough to redefine \maketag@@@:

\documentclass{article}
\usepackage{amsmath,xcolor}
\makeatletter
\def\maketag@@@#1{\hbox{\m@th\normalfont\color{black!50}#1}}
\makeatother
\begin{document}
\begin{align*}
d(x, y) &= d(y, x) \tag{symmetry} \\
&\leq d(y, z) + d(z,x) \tag{triangle}
\end{align*}
\end{document}
Note that the above would also modify the formatting of the equation numbers.
If you wish to avoid colouring references as well, then use the following:

\documentclass{article}
\usepackage{amsmath,xcolor}
\newif\ifeqref
\makeatletter
\renewcommand{\eqref}[1]{\textup{\eqreftrue\tagform@{\ref{#1}}}}
\def\maketag@@@#1{\hbox{\ifeqref\global\eqreffalse\else\color{black!50}\fi\m@th\normalfont#1}}
\makeatother
\begin{document}
\begin{align}
d(x, y) &= d(y, x) \label{abc}\\
&\leq d(y, z) + d(z,x) \tag{triangle}\label{def}
\end{align}
See~\eqref{abc} and~\eqref{def}.
\end{document}