Line break in lstlisting inside of escaped text

This segmented box is a known bug in listings package. You can use tcolorbox's listings patch to fix it. Notice that the way arguments are passed to listings package is different.

documentclass{article}
usepackage[utf8]{inputenc}
usepackage[usenames,dvipsnames,table]{xcolor}
usepackage{tcolorbox}

tcbuselibrary{skins, breakable, listings}

definecolor{keywordColor}{HTML}{0033B3}
definecolor{parameterColor}{HTML}{067D17}
definecolor{commentColor}{HTML}{8C8C8C}
definecolor{scenarioColor}{HTML}{871094}

newcommand{lstKeyword}[1]{{textcolor{keywordColor}{#1}}}
newcommand{lstParameter}[1]{{textcolor{parameterColor}{#1}}}
newcommand{lstScenario}[1]{{textcolor{scenarioColor}{textit{#1}}}}

lstdefinestyle{bddStory}
{
  captionpos=b,
  commentstyle=color{commentColor},
  keywordstyle=color{keywordColor},
  morekeywords={Given ,When ,Then ,And ,Narrative,Examples},
  escapeinside={<@}{@>},
  showspaces=false,
  showtabs=false,
  showstringspaces=false,
  breaklines=true,
  postbreak=mbox{textcolor{red}{$hookrightarrow$}space},
  breakatwhitespace=true
}

title{Example Issue}

newtcblisting{mylisting}[1]{
  enhanced,
  breakable,
  listing only,
  boxrule=0.8pt,
  sharp corners,
  top=0mm,
  bottom=0mm,
  left=2mm,
  right=2mm,
  boxsep=0mm,
  colframe=black,
  colback=white,
  listing options={
    style=#1
  }
}


begin{document}
begin{mylisting}{bddStory}
<@lstScenario{Scenario: A scenario title is too long and can't fit into a single line which causes issues in the highlighting}@>
Given there is some further content
end{mylisting}


end{document}

As for the missing arrow, I think listings package does not try to add postbreak contents to escaped region. See the example below:

If you really want to implement this feature, I find these two posts:

• Insert word, character, or symbol at beginning of all hanging indents in paragraph
• Insert a character at beginning of each line after the current one?