Duplicating Environments

When you define an environment with

\newenvironment{foo}[1]{start code with #1}{end code}

what happens internally (after some checking for optional arguments, testing if the environment exists, and taking care of a possible star argument to \newenvironment) essentially boils down to

\newcommand\foo[1]{start code with #1}% actually \new@command
\def\endfoo{end code}

This means if the environment has no optional arguments

\let\myfoo\foo
\let\endmyfoo\endfoo

copies the environment {foo} as environment {myfoo}. (If it has an optional argument \LetLtxMacro should be used instead of \let, see When to use \LetLtxMacro?.)

In the article class the {figure} environment is defined as

\newenvironment{figure}
               {\@float{figure}}
               {\end@float}

so saying

\let\myfigure\figure
\let\endmyfigure\endfigure

works:

\documentclass{article}
\newenvironment{foo}[1]{start foo with (#1)}{end foo}

\let\myfoo\foo
\let\endmyfoo\endfoo

\let\myfigure\figure
\let\endmyfigure\endfigure

\begin{document}

\begin{myfoo}{arg}
  bar
\end{myfoo}

\begin{myfigure}
  copied figure
  \caption{my figure}
\end{myfigure}

\end{document}

Some environments like {verbatim} or AMSmath's {align} cannot be copied this way since they need to find \end{verbatim} or \end{align}, respectively, exactly.

Patching \section as you do is not an especially good way to proceed, because you lose all the flexibility of the original command; one always has to look how a command is defined, before patching it. You'll discover that \section has no argument, for instance.

The same holds for environments; they are realized through a pair of commands: with

\newenvironment{foo}[1]
  {something with #1 at the start}
  {something at the end}

LaTeX defines \foo and \endfoo in a way basically equivalent to

\newcommand{\foo}[1]{something with #1 at the start}
\def\endfoo{something at the end}

In standard LaTeX the \end... command is always parameterless (and \newcommand wouldn't allow to define it).

It's also important to know when the commands are executed; here's the definition of \begin:

% latex.ltx, line 3944:
\def\begin#1{%
  \@ifundefined{#1}%
    {\def\reserved@a{\@latex@error{Environment #1 undefined}\@eha}}%
    {\def\reserved@a{\def\@currenvir{#1}%
     \edef\@currenvline{\on@line}%
     \csname #1\endcsname}}%
  \@ignorefalse
  \begingroup\@endpefalse\reserved@a}

Apart from some details, what happens with \begin{foo} is that LaTeX essentially does

\begingroup<bookkeping stuff>\foo

with some bookkeeping for later checks in the \end part. Similarly, \end{foo} will do

\endfoo<checks>\endgroup

Thus the same method used for ordinary macros will work; however,

\let\svfigure\figure
\let\endsvfigure\endfigure
\renewenvironment{figure}
  {\blahblah\begin{svfigure}}
  {\end{svfigure}\moreblah}

wouldn't be a particularly good way to patch the figure environment. For one thing, a typing mistake \end{fiugre} in the input file would result in the puzzling message

! LaTeX Error: \begin{svfigure} on input line 11 ended by \end{fiugre}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...

l.13 \end{fiugre}

and users will not understand it because they have no idea about the svfigure environment they have never started. The usual solution is to say

\let\svfigure\figure
\let\endsvfigure\endfigure
\renewenvironment{figure}
  {\blahblah\svfigure}
  {\endsvfigure\moreblah}

so that the <bookkeeping stuff> will be done for figure but not for the inner svfigure.

However, not all environments are happily redefined this way.

Environments taking an optional argument should use \LetLtxMacro (with \usepackage{letltxmacro}) instead of \let, but this is true also for macros. For the \end... command \let is always sufficient. More importantly, this way of doing might not work with environments defined with different tools, notably \NewEnviron (environ package) or \NewDocumentEnvironment (xparse package).

Some other environments have their own peculiarities: don't try redefining lrbox (which wouldn't make sense anyway), nor verbatim. Something like

\let\svverbatim\verbatim
\let\endsvverbatim\endverbatim
\renewenvironment{verbatim}
  {<something>\svverbatim}
  {\endsvverbatim<something>}

will simply not work and keep you in permanent verbatim mode unless you load the verbatim package. But don't try this for changing the font size in verbatim, it won't work.

Patching macros and environments is sometimes an arcane activity best suited for necromancers rather than even skilled LaTeX programmers.