Keep at all times, the minus sign above aligned with minus sign below

Do you mean like this?

\documentclass{article}
\usepackage{xparse,array}

\ExplSyntaxOn

\NewDocumentCommand{\frg}{mm}
 {
  \dbn_frg:nn { #1 } { #2 }
 }

\tl_new:N \l__dbn_frg_num_tl
\tl_new:N \l__dbn_frg_den_tl

\cs_new_protected:Nn \dbn_frg:nn
 {
  \tl_set:Nn \l__dbn_frg_num_tl { #1 }
  \tl_set:Nn \l__dbn_frg_den_tl { #2 }
  \regex_replace_once:nnN { (\+|\-) } { \cT\& \1 \cT\& } \l__dbn_frg_num_tl
  \regex_replace_once:nnN { (\+|\-) } { \cT\& \1 \cT\& } \l__dbn_frg_den_tl
  \begin{array}{@{} r @{} >{{}}c<{{}} @{} l @{} }
  \l__dbn_frg_num_tl \\
  \hline
  \l__dbn_frg_den_tl
  \end{array}
 }

\ExplSyntaxOff

\begin{document}

\[
\frg{4-3}{4-1}\qquad \frg{4+3}{4+11} \qquad \frg{15-2}{6+12}
\]

\end{document}

I do not know what the purpose of this exercise is but alignment can e.g. achieved with an array. Notice that I didn't pay any attention to spacing, bold and etc. because I do not know the purpose.

\documentclass{article}
\makeatletter
\def\frg@aux#1-#2,#3-#4\@nil{\begin{array}{@{\,}r@{}c@{}l@{\,}}
#1&-&#2\\
\hline
#3&-&#4
\end{array}}
\def\frg#1{\frg@aux#1\@nil}
\makeatother

\begin{document}
\[ a=\frg{4 - 3,  3 - 1  }\]
\end{document}