Modify only the current system

Starting bars usually are all clones of the first one. The only way to affect one system and not the others is to collect all the grobs using the all-elements property of any grob on the system in question (for example a NoteHead) and filter the target grob, which you can then modify.

\version "2.24.0"

%% Thanks to David Nalesnik:
%% https://lists.gnu.org/archive/html/lilypond-user/2019-03/msg00205.html

#(define drop-in-on-spanner
   (lambda (grob)
     (let* ((elts (ly:grob-array->list (ly:grob-object (ly:grob-system grob) 'all-elements)))
            (ssb (filter (lambda (e) (grob::has-interface e 'system-start-delimiter-interface))
                         elts)))
       (set! (ly:grob-property (car ssb) 'color) red))))

\new StaffGroup <<
  \new Staff {
    c'1 \break
    \tweak after-line-breaking #drop-in-on-spanner
    c' \break
    c' \break
  }
  \new Staff {
    c' c' c'
  }
>>

\paper { tagline = ##f }