Align position of a table exactly to another one

I don't see why you want to use floatrow. However, here's a possibility: measure the first table and set the second one in a minipage as wide. The usage of geometry is just for drawing a frame around the page to see the placement with respect to the margins.

\documentclass[a4paper]{scrreprt}
\usepackage[pass,showframe]{geometry} % just for this example
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{booktabs}
\usepackage[labelfont=bf, labelsep=newline,singlelinecheck=false]{caption}
\usepackage{floatrow}

\newbox{\firsttabbox}

\KOMAoption{captions}{tableheading}
\setcapindent{0em}
\pagestyle{headings}

\begin{document}

\begin{table}[htpb]
\sbox{\firsttabbox}{\begin{tabular}{lccccc}
\toprule
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE  \\
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE  \\
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE \\
\bottomrule
\end{tabular}}
\begin{floatrow}
\ttabbox
  {\usebox{\firsttabbox}}
  {\caption{ABCDE table}\label{tab:ABCDE}}
\end{floatrow}

\bigskip

\begin{floatrow}
\ttabbox{%
  \begin{minipage}{\wd\firsttabbox}
  \raggedright
  \begin{tabular}{lccc}
  \toprule
  ABCDE & ABCDE & ABCDE & ABCDE \\
  ABCDE & ABCDE & ABCDE & ABCDE \\
  ABCDE & ABCDE & ABCDE & ABCDE \\
  \bottomrule
  \end{tabular}
  \end{minipage}}
  {\caption{ABCDE table}\label{tab:ABCD}}
\end{floatrow}
\end{table}


\end{document}

When you have set the save box with \sbox{\firsttabbox}, its width is available as \wd\firsttabbox. So we can feed this width to the following minipage and “use” the saved box in the first floatrow.

Pay attention that your usage

\tabbox{
  <material>
}
{
 \caption{...}
}

adds spaces which should be protected with % or with other tokens. So if for instance you want the second \tabbox with “isolated” braces, you must do

\ttabbox{% <-- important
  \begin{minipage}{\wd\firsttabbox}
  \raggedright
  \begin{tabular}{lccc}
  \toprule
  ABCDE & ABCDE & ABCDE & ABCDE \\
  ABCDE & ABCDE & ABCDE & ABCDE \\
  ABCDE & ABCDE & ABCDE & ABCDE \\
  \bottomrule
  \end{tabular}
  \end{minipage}% <-- important
}
{% <-- important
 \caption{ABCDE table}\label{tab:ABCD}% <-- important
}

A % after \end{tabular} is not needed, because \end{minipage} gets rid of that space.

---

A solution without floatrow; we first typeset the tables for knowing their width and then do the final typesetting.

\documentclass[a4paper]{scrreprt}
\usepackage[pass,showframe]{geometry} % just for this example
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{booktabs}
\usepackage[labelfont=bf, labelsep=newline,singlelinecheck=false]{caption}
\usepackage{environ}

\KOMAoption{captions}{tableheading}
\setcapindent{0em}
\pagestyle{headings}

\NewEnviron{lefttables}[1][tp]{%
  \global\tabwidth=0pt
  \vbox{
    \renewcommand\caption[2][]{}%
    \renewcommand\label[1]{}%
    \let\latextabular\tabular
    \let\latexendtabular\endtabular
    \def\tabular{\setbox8=\hbox\bgroup\latextabular}%
    \def\endtabular{\latexendtabular\egroup\settabwidth}
    \BODY
  }
  \begin{table}[#1]
  \centering
  \begin{minipage}{\tabwidth}\raggedright
  \BODY
  \end{minipage}
  \end{table}
}
\newdimen\tabwidth
\newcommand{\settabwidth}{%
  \ifdim\wd8>\tabwidth \global\tabwidth=\wd8 \fi
}


\begin{document}

\begin{lefttables}[htpb]
\caption{ABCDE table}\label{tab:ABCDE}

\begin{tabular}{lccccc}
\toprule
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE  \\
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE  \\
ABCDE & ABCDE & ABCDE & ABCDE & ABCDE & ABCDE \\
\bottomrule
\end{tabular}

\bigskip

\begin{tabular}{lccc}
\toprule
ABCDE & ABCDE & ABCDE & ABCDE \\
ABCDE & ABCDE & ABCDE & ABCDE \\
ABCDE & ABCDE & ABCDE & ABCDE \\
\bottomrule
\end{tabular}
\end{lefttables}

\end{document}