Expl3 and environ to fill boxes title, upper part, and lower part from only user text

This does something of the sort. (I prefer the cleaner, traditional defs, but you can convert this to expl3 if you need.) This code comes with a variant of your bubble box that allows for a tcblower.

documentclass{article}
usepackage[many]{tcolorbox}
usepackage{varwidth}
usepackage{environ}
usepackage{xparse}

newlength{bubblesep}
newlength{bubblewidth}
setlength{bubblesep}{2pt}
AtBeginDocument{setlength{bubblewidth}{.75textwidth}}

newcommand{bubble}[2][]{%
  tcbox[
    on line,
    arc=2.5mm,
    rounded corners,
    title=#1,halign title=right
  ]{begin{varwidth}{bubblewidth}#2end{varwidth}}%
}

newcommand{bubbledual}[3][]{%
setbox0hbox{begin{varwidth}{bubblewidth}
small #2
end{varwidth}}%
setbox1hbox{begin{varwidth}{bubblewidth}
#3
end{varwidth}}%
ifdimwd0>wd1relax
edefmybubblewidth{thedimexprwd0+1.2cm}%
else
edefmybubblewidth{thedimexprwd1+1.2cm}%
fi
  begin{tcolorbox}[width=mybubblewidth,
    arc=2.5mm,
    rounded corners,
    title=#1,halign title=right
  ]
  begin{varwidth}{bubblewidth}
  {small #2}
  end{varwidth}%
  tcblower
  begin{varwidth}{bubblewidth}
  #3
  end{varwidth}%
end{tcolorbox}}

defbubblesplit#1newline#2newline{ifx#2relaxrelax
bubble[mytitle]{#1}%
else
bubbledual[mytitle]{#1}{#2}%
fi
}%



ExplSyntaxOn
seq_new:N l__itt_bubbles_seq
tl_new:N l__itt_bubbles_first_tl
tl_new:N l__itt_bubbles_last_tl

NewEnviron{rightbubbles}[1][]
 {
  begin{flushright}
  sffamily
  seq_set_split:NnV l__itt_bubbles_seq { par } BODY
  int_compare:nTF { seq_count:N l__itt_bubbles_seq < 2 }
   {
     %bubble[#1]{BODY}par
      defmytitle{#1}%
      letoldnewlinenewline
      letnewlinerelax
      expandafterbubblesplit##1newlinenewline
      letnewlineoldnewline
   }
   {
    seq_map_inline:Nn l__itt_bubbles_seq
     {
      %bubble[#1]{##1}
      defmytitle{#1}%
      letoldnewlinenewline
      letnewlinerelax
      expandafterbubblesplit##1newlinenewline
      letnewlineoldnewline
      parnointerlineskip
      addvspace{bubblesep}
     }
   }
  end{flushright}
 }
ExplSyntaxOff

begin{document}

begin{rightbubbles}[ member A ]
20:20:11" 26 nov. 2020newline
Right-aligned gray bubbles with black text

20:23:12" 26 nov. 2020newline
Bubbles only break after a paragraph (equivalent to an enter press 
when chatting). Long message with multiple lines will be kept in one bubble.

20:27:54" 26 nov. 2020newline
Left and right edges are round.

20:30:06" 26 nov. 2020newline
Single line dots

a line without time stamp
end{rightbubbles}

end{document}