Why does this language test fail?

For implementation reasons, the replacement text of \languagename has its first character of category code 12; as far as \ifx is concerned,

e₁₁n₁₁g₁₁l₁₁i₁₁s₁₁h₁₁

and

e₁₂n₁₁g₁₁l₁₁i₁₁s₁₁h₁₁

are different. Indeed, if you do

\documentclass{article}
\usepackage[english]{babel}
\begin{document}
\edef\suplang{\string english}
\ifx\languagename\suplang
    Supported language: \languagename
\else
    Not supported language: \languagename
\fi
\end{document}

then the output is

Do you need to do that business? No, of course not.

\usepackage{pdftexcmds} % to work with all engines
\makeatletter
\newcommand{\supported}[3]{%
  \ifnum\pdf@strcmp{\languagename}{#1}=\z@
     \expandafter\@firstoftwo
  \else
     \expandafter\@secondoftwo
  \fi
  {#2}{#3}%
}
\makeatother

Then

\supported{suplang}
  {Supported language: \languagename}
  {Not supported language: \languagename}

will work.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{pdftexcmds} % to work with all engines
\makeatletter
\newcommand{\supported}[3]{%
  \ifnum\pdf@strcmp{\languagename}{#1}=\z@
     \expandafter\@firstoftwo
  \else
     \expandafter\@secondoftwo
  \fi
  {#2}{#3}%
}
\makeatother

\def\suplang{english}

\begin{document}

\supported{\suplang}
  {Supported language: \languagename}
  {Not supported language: \languagename}

\end{document}

Alternative definition with \else and \fi for the true and false branches:

\documentclass{article}
\usepackage[english]{babel}
\usepackage{pdftexcmds} % to work with all engines
\makeatletter
\newcommand{\issupported}[1]{%
  TT\fi\ifnum\pdf@strcmp{\languagename}{#1}=\z@
}
\makeatother

\def\suplang{english}

\begin{document}

\if\issupported\suplang
  Supported language: \languagename
\else
  Not supported language: \languagename
\fi

\end{document}

The answer of egreg already explains the reason for the unexpected result of the example in the question and also shows several methods to solve the issue.

Package iflang wraps the internal stuff of the comparison with \languagename in a simple macro \IfLanguageName. The example of the question becomes:

\documentclass{article}
\usepackage[english]{babel}
\usepackage{iflang}
\begin{document}
\IfLanguageName{english}{%
    Supported language: \languagename
}{%
    Not supported language: \languagename
}
\end{document}