The following suggestions only work if you center both pictures.
You can use the option trim axis left (or maybe trim axis right) for both tikzpictures:
\begin{tikzpicture}[trim axis left]

Code:
\documentclass{scrartcl}
\usepackage{pgfplots}
\pgfplotsset{width=8cm,height=6cm}
\begin{document}
\begin{center}
\begin{tikzpicture}[trim axis left]
\begin{axis}[no marks]
\addplot {x};
\end{axis}
\end{tikzpicture}
\captionof{figure}{Fugue 1}
\vspace{2\baselineskip}
\begin{tikzpicture}[trim axis left]
\begin{axis}[ylabel = {$y(x)$},no marks]
\addplot {2*x};
\end{axis}
\end{tikzpicture}
\captionof{figure}{Figure 2}
\end{center}
\end{document}
Another possibility is the use of \pgfresetboundingbox and afterwards \useasboundingbox
\documentclass{scrartcl}
\usepackage{pgfplots}
\pgfplotsset{width=8cm,height=6cm}
\newcommand\myBoundingBox{
\pgfresetboundingbox
\path(current axis.below south west)rectangle (current axis.above north east);
}
\begin{document}
\begin{center}
\begin{tikzpicture}
\begin{axis}[no marks]
\addplot {x};
\end{axis}
\myBoundingBox
\end{tikzpicture}
\captionof{figure}{Fugue 1}
\vspace{2\baselineskip}
\begin{tikzpicture}
\begin{axis}[ylabel = {$y(x)$},no marks]
\addplot {2*x};
\end{axis}
\myBoundingBox
\end{tikzpicture}
\captionof{figure}{Figure 2}
\end{center}
\end{document}
Or you can use the environment pgfinterruptboundingbox for the axis and afterwards \useasboundingbox
...
\begin{pgfinterruptboundingbox}
\begin{axis}[...
...
\end{axis}
\end{pgfinterruptboundingbox}
\useasboundingbox(current axis.below south west)rectangle (current axis.above north east);
...
\centeringso without more detail it's hard to say anything about your question. – David Carlisle Jul 12 '14 at 09:31