How to make equal sign with circles, similar to \risingdotseq?

Here's a not-so-simple™ solution:

\documentclass{article}
\usepackage{amsmath,pict2e}

\makeatletter
\newcommand{\raisingcircleseq}{\mathrel{\mathpalette\raising@circles@eq\relax}}
\newcommand{\raising@circles@eq}[2]{%
  \vphantom{#1+}%
  \vbox{
    \settowidth\unitlength{$#1\mspace{2mu}$}%
    \offinterlineskip\m@th
    \ialign{##\cr
      \hfil\small@circle{#1}$#1\mspace{1.5mu}$\cr\noalign{\vskip0.5\unitlength}
      $#1=$\cr\noalign{\post@vskip{+}{#1}}
      $#1\mspace{1.5mu}$\small@circle{#1}\hfill\cr\noalign{\post@vskip{-}{#1}}
    }%
  }%
}
\newcommand{\small@circle}[1]{%
  \smash{%
    \begin{picture}(1,1)
    \small@linethickness{#1}
    \put(0.5,0.5){\circle{1}}
    \end{picture}%
  }%
}
\newcommand{\small@linethickness}[1]{%
  \linethickness{%
      \ifx#1\displaystyle 0.8\fontdimen8\textfont3\else
      \ifx#1\textstyle 0.8\fontdimen8\textfont3\else
      \ifx#1\scriptstyle0.8\fontdimen8\scriptfont3\else
      1\fontdimen8\scriptscriptfont3\fi\fi\fi
  }%
}
\newcommand{\post@vskip}[2]{%
  \expandafter\vskip\expanded{%
    #1\ifx#2\scriptscriptstyle0.9\else\ifx#2\scriptstyle0.6\else0.3\fi\fi\unitlength
  }%
}
\makeatother

\begin{document}

$a=\raisingcircleseq b$

$a\raisingcircleseq b$

${=\raisingcircleseq}{\scriptstyle=\raisingcircleseq}{\scriptscriptstyle=\raisingcircleseq}$

\end{document}

Here's an option.

\documentclass{article}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\newlength{\circheight}
\settoheight{\circheight}{\(=\)}
\addtolength{\circheight}{.5pt}
\newcommand*{\smallcirc}{\scalebox{.51}{\(\scriptscriptstyle\boldsymbol{\circ}\)}}
\newcommand*{\risingcircleseq}{%
    \mathrel{%
        \makebox[0pt][l]{\raisebox{-.5pt}{\smallcirc}}%
        \mbox{=}%
        \makebox[0pt][r]{\raisebox{\circheight}[0pt]{\smallcirc}}%
    }%
}
\begin{document}
\(A \risingcircleseq B\)
\end{document}

Works in all math styles. Uses stackengine to build the macro and scalerel for automatic handling of math style.

\documentclass{article}
\usepackage{stackengine,graphicx,scalerel}
\newcommand\eqdots{\mathrel{\ensurestackMath{\ThisStyle{%
  \stackengine{-.5\LMpt}{\stackengine{.5\LMpt}{\SavedStyle=}%
  {\SavedStyle\scaleobj{.33}{\circ\,}}{O}{r}{F}{T}{S}}%
  {\SavedStyle\scaleobj{.33}{\,\circ}}{U}{l}{F}{T}{S}}}}}
\begin{document} 
$x\eqdots y$\par
$\scriptstyle x\eqdots y$\par
$\scriptscriptstyle x\eqdots y$\par
\end{document}