Reuse a part of a tikz picture with \savebox

A \savebox can only store a complete box. A complete tikzpicture qualifies for being used in a \savebox, a collection of paths does not. The TikZy way of doing this is to define a pic. (style=grid wasn't defined in your MWE so I made something up.)

\documentclass[tikz]{standalone}
\tikzset{grid/.style={draw=gray,thin},
pics/alwaysthesame/.style={code={%
\draw[style=grid] (-0.99,-0.99) grid (8.99,8.99);
\draw[thick,->] (-1,0) -- (9,0) node[pos=0.97,below] {$x$};
\draw[thick,->] (0,-1) -- (0,9) node[pos=0.97,left] {$y$};
}}}
\begin{document}

\begin{tikzpicture}
\pic{alwaysthesame};
% Graph 1 
\draw[very thick,color=black,domain=-1:9] plot (\x,{\x});
\end{tikzpicture}

\begin{tikzpicture}
\pic{alwaysthesame};
% Graph 2 
\draw[very thick,color=black,domain=-1:9] plot (\x,{0.5*\x});
\end{tikzpicture}

\end{document}

You can also use the every picture and execute at begin picture keys to spare you from punching in the pics in every picture. Now with the updated code for the grid.

\documentclass[tikz]{standalone}
\tikzset{pics/alwaysthesame/.style={code={%
\draw[step=1,help lines,black!20] (-0.99,-0.99) grid (8.99,8.99);
\draw[thick,->] (-1,0) -- (9,0) node[pos=0.97,below] {$x$};
\draw[thick,->] (0,-1) -- (0,9) node[pos=0.97,left] {$y$};
}},every picture/.append style={execute at begin picture={\pic{alwaysthesame};}}}
\begin{document}

\begin{tikzpicture}
% Graph 1 
\draw[very thick,color=black,domain=-1:9] plot (\x,{\x});
\end{tikzpicture}

\begin{tikzpicture}
% Graph 2 
\draw[very thick,color=black,domain=-1:9] plot (\x,{0.5*\x});
\end{tikzpicture}

\end{document}

You could save a complete tikzpicture in a \savebox and put it in a node. This does work, but you need to be very careful to use e.g. inner sep when you do that.

\documentclass[tikz]{standalone}
\newsavebox\AlwaysTheSame
\sbox\AlwaysTheSame{\begin{tikzpicture}
\draw[step=1,help lines,black!20] (-0.99,-0.99) grid (8.99,8.99);
\draw[thick,->] (-1,0) -- (9,0) node[pos=0.97,below] {$x$};
\draw[thick,->] (0,-1) -- (0,9) node[pos=0.97,left] {$y$};
\end{tikzpicture}}
\begin{document}

\begin{tikzpicture}
\node[anchor=south west,inner sep=0pt,outer sep=0pt] at (-0.99,-0.99)
{\usebox\AlwaysTheSame};
% Graph 1 
\draw[very thick,color=black,domain=-1:9] plot (\x,{\x});
\end{tikzpicture}

\begin{tikzpicture}
\node[anchor=south west,inner sep=0pt,outer sep=0pt] at (-0.99,-0.99)
{\usebox\AlwaysTheSame};
% Graph 2 
\draw[very thick,color=black,domain=-1:9] plot (\x,{0.5*\x});
\end{tikzpicture}

\end{document}

In addition to the solution Schrodinger's cat provided, it's also possible to do this using \newcommand.

Someone should explain here why boxes are not the right way to approach this problem. I don't think this is the clearest explanation, but here it is. Roughly speaking, a \box is a collection of stored material that has already been formatted. When you "use" a box, the already formatted material stored in the box is just dumped onto the page. In particular, no logic is allowed to be stored in the box. Tikz needs a whole lot of logic in order to control the drawing global state (so for instance, if tikz adds a curve to the pdf, the color, line thickness, and dash pattern used to draw the curve depends on the global state). So if you want to store up Tikz code, store it in a macro or using one of the techniques Scrodinger's cat indicated.

\documentclass[tikz]{standalone}

\begin{document}

\tikzset{gridstyle/.style={help lines}}
\newcommand{\mygrid}{
    \draw[gridstyle] (-0.99,-0.99) grid (8.99,8.99);
    \draw[thick,->] (-1,0) -- (9,0) node[pos=0.97,below] {$x$};
    \draw[thick,->] (0,-1) -- (0,9) node[pos=0.97,left] {$y$};
}

\begin{tikzpicture}
\mygrid
% Graph 1
\draw[very thick,color=black,domain=-1:9] plot (\x,{\x});
\end{tikzpicture}

\begin{tikzpicture}
\mygrid
% Graph 2
\draw[very thick,color=black,domain=-1:9] plot (\x,{0.5*\x});
\end{tikzpicture}
\end{document}