The error has nothing to do with eqnarray (which should not be used anyway) apart from it being an alignment environment.
The problem is that tabstackengine doesn't protect the internal & from a possible outer alignment.
Just brace the matrix:
\documentclass{report}
\usepackage{amsmath}
\usepackage{tabstackengine}
\begin{document}
\begin{align*}
v &= {\bracketMatrixstack{1 & 2 \\ 3 & 3}} \bracketVectorstack{-3 \\ 2} \\
a &= b
\end{align*}
\end{document}

I'm not sure how this is better than
\documentclass{report}
\usepackage{amsmath}
\begin{document}
\begin{align*}
v &= \begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}
\begin{bmatrix} -3 \\ 2 \end{bmatrix} \\
a &= b
\end{align*}
\end{document}

You can fix tabstackengine:
\documentclass{report}
\usepackage{amsmath}
\usepackage{tabstackengine}
\makeatletter
\renewcommand\ensureTABstackMath[1]{%
{\ifnum`}=\z@\fi
\let\sv@TABmode\TAB@delim\TABstackMath#1\let\TAB@delim\sv@TABmode
\ifnum`{=\z@\fi}%
}
\makeatother
\begin{document}
\begin{align*}
v &= \bracketMatrixstack{1 & 2 \\ 3 & 3} \bracketVectorstack{-3 \\ 2} \\
v &= \begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}
\begin{bmatrix} -3 \\ 2 \end{bmatrix} \\
a &= b
\end{align*}
\end{document}
but I still can't see any improvement.

eqnarraryshouldn't be used any longer any way – Feb 28 '18 at 17:08tabstackengine 2018/03/05 (V2.10)was released earlier this week that fixes this issue and adds some other bells and whistles. – Steven B. Segletes Mar 07 '18 at 15:56