Add ‘Chapter’ beginning on the Title in the Second Language Content

The tocloft package can help, read the documentation (texdoc tocloft).

After the end of your English Contents add

usepackage{tocloft}
renewcommand{cftchappresnum}{Chapter } % put Chapter before number
addtolength{cftchapnumwidth}{4em} % Extra space for Chapter

This worked with my rendering of your code into an MWE (shown below) which of course didn't process maketableofcontents as you had shown no code for that. I have no idea about your Chinese chapter titles if they will be affected.

% engchintocprob.tex  SE 539522

documentclass{report}

%—————————English Contents—————————————
makeatletter
newcommandengcontentsname{Contents}
newcommandtableofengcontents{%
    if@twocolumn
      @restonecoltrueonecolumn
    else
      @restonecolfalse
    fi
    chapter*{engcontentsname
        @mkboth{%
           MakeUppercaseengcontentsname}{MakeUppercaseengcontentsname}}%
    @starttoc{toe}% !!!!Define a new contents!!!!
    if@restonecoltwocolumnfi
    }
newcommandaddengcontents[2]{%
    addcontentsline{toe}{#1}{protectnumberline{csname the#1endcsname}#2}}
makeatother

newcommandechapter[1]{addengcontents{chapter}{#1}}
newcommandesection[1]{addengcontents{section}{#1}}
newcommandesubsection[1]{addengcontents{subsection}{#1}}
newcommandesubsubsection[1]{addengcontents{subsubsection}{#1}}
%—————————English Contents—————————————

usepackage{tocloft}
renewcommand{cftchappresnum}{Chapter }
addtolength{cftchapnumwidth}{4em}

begin{document}
pagenumbering{Roman}setcounter{page}{1}
tableofcontents
maketableofcontents  %Chinese 

tableofengcontents    %English 
cleardoublepage
echapter{First}

end{document}