How to get rid of & in cases environment but still get the same space in the surroundings

You could use aligned

\documentclass{article}
\usepackage{amsmath}

\newenvironment{system}
  {\[\left\lbrace\aligned}
  {\endaligned\right.\]\ignorespacesafterend}

\begin{document}
Example:
\begin{system}
&x+y=2\\&x=2
\end{system}


\end{document}

Remove the intercolumn space and also the null delimiter space.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newenvironment{system}
 {%
  \let\@ifnextchar\new@ifnextchar
  \left\lbrace
  \def\arraystretch{1.2}%
  \array{@{}l@{}}%
 }
 {%
  \endarray
  \right.\kern-\nulldelimiterspace
 }
\makeatother

\begin{document}

\[
\begin{system}
x+y=2,\\x=2
\end{system}
\equiv
\begin{system}
2+y=2,\\x=2
\end{system}
\]

\end{document}

I have doubts your readers will understand what you mean here.

\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage[usestackEOL]{stackengine}
\stackMath
\setstackgap{L}{16pt}
\begin{document}
\[
\biggl\{\Centerstack[l]{x+y=2,\\x=2}
\equiv
\biggl\{\Centerstack[l]{2+y=2,\\x=2}
\]
\end{document}