How to draw tikz node behind text

TikZ may be an overkill for such simple tasks, you can try tcolorbox, mdframed, fancybox, etc. If you still like TikZ, your example can be easily modified.

\documentclass[12pt,a4paper]{book}
\usepackage{tikz,amsmath}
\usepackage{graphicx}
\begin{document}

\begin{equation*}
  y(x) = g(x)\,
  \tikz[remember picture, overlay]{
    \node[fill=gray!20, inner sep=1pt, anchor=text, rectangle, rounded corners=1mm]{$f(x)$};
  }
  \phantom{f(x)}\,
  h(x)
\end{equation*}

\end{document}

This answer is quite complicated, but it achieves the desired effect of keeping the exact spacing as before but with the gray box in the background:

\documentclass{article}

\usepackage{amsmath}
\usepackage{tikz}

\pgfdeclarelayer{bg}
\pgfsetlayers{bg,main}

\begin{document}
    \begin{equation*}
        y(x) = 
        %% remember position of g(x), but don't draw it yet:
        \tikz[remember picture,baseline=(gx.base)]{
            \node [inner sep=0,anchor=text] (gx) {\phantom{$g(x)$}};
        }
        %% draw f(x) and g(x) in the correct order:
        \tikz[remember picture,baseline=(fx.base)]{
            \node [inner sep=0,anchor=text] (fx) {$f(x)$}; %% draw f(x)
            %% do not take up any space for the box or g(x), as we already have that:
            \begin{pgfinterruptboundingbox}
                \begin{pgfonlayer}{bg} %% draw on the background layer
                    %% draw the gray box:
                    \node at (fx.center) [anchor=center] %% exact position as f(x)
                        [fill=gray!20, inner sep=2pt,  rectangle, rounded corners=1mm]
                        {\phantom{$f(x)$}} %% exact same size as f(x)
                        ;
                    %% draw g(x):
                    \node at (gx.center) [anchor=center] %% exact position of g(x)
                        {$g(x)$};
                \end{pgfonlayer}
            \end{pgfinterruptboundingbox}
        }
        h(x)
    \end{equation*}
\end{document}

Here is another answer with the exactly correct spacing. In contrast to my other answer, this one doesn't need layers.

\documentclass[border={0pt 2pt 0pt 2pt}]{standalone}

\usepackage{tikz}

\begin{document}
    $
        y(x) = 
        %% remember position of g(x), but don't draw it yet:
        \tikz[remember picture,baseline=(gx.base)]{
            \node [inner sep=0,anchor=text] (gx) {\phantom{$g(x)$}};
        }
        %% draw f(x) and g(x) in the correct order:
        \tikz[remember picture,baseline=(fx.base)]{
            %% remember position of g(x), but don't draw it yet:
            \node [inner sep=0,anchor=text] (fx) {\phantom{$f(x)$}};
        }
        %% Now draw everything in the correct order:
        \tikz[remember picture,overlay]{ %% overlay, because the space is already taken by the \phantoms
            %% draw the gray box:
            \node at (fx.center) [anchor=center] %% exact position as f(x)
                [fill=gray!20, inner sep=2pt,  rectangle, rounded corners=1mm]
                {$f(x)$} %% exact same size as f(x)
                ;
            %% draw g(x):
            \node at (gx.center) [anchor=center] %% exact position of g(x)
                {$g(x)$};
        }
        h(x)
    $
\end{document}