Why is there multiply labels in this gather environment?

It's a known problem, see https://github.com/latex3/latex2e/issues/15

The manual of amsmath explicitly tells that align can be nested in gather, see page 26:

However a “register” is not cleared at the proper time and a spurious, but innocuous, warning of There were multiply-defined labels is issued; this happens in connection with \nonumber.

In my experiment, patching \endalign solves the issue.

\documentclass[11pt,letterpaper,twoside]{book}
\usepackage[total={6in,10in},left=1.5in,top=0.5in,includehead,includefoot]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{xpatch}

\makeatletter
\xpatchcmd{\endalign}
 {\restorealignstate@}
 {\global\let\df@label\@empty\restorealignstate@}
 {}{}
\makeatother

\begin{document}

TEST
\begin{gather}
    A \equiv X + Y \nonumber \\
\begin{align}
    &= C + D + E + F + G \nonumber \\
    &= E. \label{some label}
\end{align}
\end{gather}

\end{document}

I do not think one should put an align environment into a gather environment, but you can use aligned. I think that gather and align should not be nested as they start their own math environment, but aligned is precisely made for having an align type math environment inside another math environment.

\documentclass[11pt,letterpaper,twoside]{book}
\usepackage[total={6in,10in},left=1.5in,top=0.5in,includehead,includefoot]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}

\begin{document}

TEST
\begin{gather}
    A \equiv X + Y \nonumber \\
\begin{aligned}[b]\label{some label}
    &= C + D + E + F + G \\
    &= E. 
\end{aligned}
\end{gather}

ANOTHER TEST
\begin{align}
\MoveEqLeft    A \equiv X + Y \notag \\
    &= C + D + E + F + G \notag\\
    &= E. \label{some label}
\end{align}

\end{document}