- in equation environment had not to be empty lines
- it is not entirely unclear, what you mean with centering.
Is the following simple solution close to what you after?

(red lines shows text borders)
\documentclass{article}
\usepackage{tikz-cd}
%---------------- Show page layout. Don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%
\begin{document}
\[
\begin{tikzcd}[column sep=large]
\ast\times X\arrow[r,"{(1,1_X)}"]\arrow[dr,swap,"\cong"] & G\times X\arrow[d,"\alpha"]\\
&X
\end{tikzcd}
%
\qquad
%
\begin{tikzcd}[column sep=large]
G\times G\times X\arrow[r,"{(m,1_X)}"]\arrow[d,swap,"{(1_G,\alpha )}"] & G\times
X\arrow[d,"\alpha"]\\
G\times X \arrow[r,"\alpha"]&X
\end{tikzcd}
\]
\end{document}
Addendum:
It seems that OP looking for the following:

(red lines indicate text borders,
vertical black lines show positions of image bisectors)
\documentclass{article}
\usepackage{tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\usepackage{tikz-cd}
%---------------- Show page layout. Don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%
\begin{document}
\noindent\begin{tabularx}{\linewidth}{ C | C } % in real document delete `|`
\begin{tikzcd}[column sep=large,
ampersand replacement=\&]
\ast\times X\arrow[r,"{(1,1_X)}"]\arrow[dr,swap,"\cong"]
\& G\times X\arrow[d,"\alpha"]\\
\& X
\end{tikzcd}
& \begin{tikzcd}[column sep=large,
ampersand replacement=\&]
G\times G\times X\arrow[r,"{(m,1_X)}"]\arrow[d,swap,"{(1_G,\alpha )}"]
\& G\times
X\arrow[d,"\alpha"]\\
G\times X \arrow[r,"\alpha"]
\& X
\end{tikzcd} \\
\vline & \vline % in real document delete this code line
\end{tabularx}
\end{document}