Centering in TikZ matrix of nodes

align=center without any text width is the problem. Comment it out and all text will be centered (with or without space after it) or add a text width over which the text could be centered.

\documentclass[tikz]{standalone}
\usetikzlibrary{matrix}
\begin{document}
    \begin{tikzpicture}
        \matrix [
            matrix of nodes,
            row sep=-\pgflinewidth,
            column sep=-\pgflinewidth,
            nodes={
                rectangle, draw=black, minimum height=1.25em, minimum width=1.25em,
                anchor=center, %align=center, %text width=2em,
                inner sep=0pt, outer sep=0pt
            }
        ] {
            0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 \\
        };
    \end{tikzpicture}
\end{document}

I am not sure what other bells and whistles you want in your real application but rather than using a matrix of nodes for this you might want to consider using a macro with an embedded for-loop:

\documentclass[tikz, border=4mm]{standalone}

\newcommand\myarray[1]{%
  \begin{tikzpicture}
    \foreach \x [count=\c] in {#1} {
       \draw[thick](\c-0.5,0-0.5) rectangle ++ (1,1);
       \draw(\c,0)node{\x};
    }
  \end{tikzpicture}%
}

\begin{document}
   \myarray{0, 3, 6, 9, 12, 15, 18, 21}
\end{document}

This produces:

The main benefit of this approach is that it simplifies the code quite a lot, especially if you have a lot of these arrays. Of course, you can fine tune at will with shading, colours, font sizes, box size etc etc.

EDIT

Motivated by the OP's comment, here is a 2D version:

\documentclass[tikz, border=4mm]{standalone}

\newcommand\myarray[1]{%
  \begin{tikzpicture}
    \foreach \row [count=\r] in {#1} {
      \foreach \x [count=\c] in \row {
        \draw[thick](\c-0.5,-\r-0.5) rectangle ++ (1,1);
        \draw(\c,-\r)node{\x};
       }
     }
  \end{tikzpicture}%
}

\begin{document}
   \myarray{{0,3,6,9,12,15,18,21},{2,4,7,11},{4,5,6,7}}
\end{document}

This code produces: