How to draw a stack in drawstack/TiKz?

TiKZ could be an alternative to drawstack.

\documentclass[tikz, border=2mm]{standalone}
\usetikzlibrary{matrix,positioning}
\usepackage{lmodern}
\usepackage{fontawesome5}


\begin{document}
\begin{tikzpicture}[
    level/.style={draw, minimum width=3cm, minimum height=8mm},
    stack/.style={matrix of nodes, nodes={level}, row sep=-\pgflinewidth},
    ]

    \matrix[stack, label={[font=\small, align=center, name=aux1]below:{departure\\ airport}},
    label={[font=\Large, name=p1]above:\faPlaneDeparture}] (stackleft){
        ticket (purchase) \\
        baggage (check) \\
        gates (load) \\
        runway (takeoff) \\
        airplane routing \\};

    \node[level, right= of stackleft-5-1] (ar1) {airplane routing};
    \node[level, right=of ar1] (ar2) {airplane routing};

    \matrix[stack, label={[font=\small, align=center, name=aux2]below:{arrival\\ airport}}, 
    label={[font=\Large, name=p2]above:\faPlaneArrival},
        right=of ar2, anchor=stackright-5-1.west ] (stackright){
        ticket (complain) \\
        baggage (claim) \\
        gates (unload) \\
        runway (land) \\
        airplane routing \\};

    \node[font=\Large] at (p1-|ar1) {\faPlane};
    \node[font=\Large] at (p1-|ar2) {\faPlane};

    \path (aux1)--node[align=center, font=\small]{intermediate air-traffic\\ control centers} (aux2);

    \draw[blue, very thick, ->] (stackleft.north west)--(stackleft.west|-aux1)--(aux1.south)--(aux2.south)--(stackright.east|-aux2)--(stackright.north east);
\end{tikzpicture}
\end{document}

drawstack is poorly document package and consequently not very useful for your needs. better is to use pure tikz. with multi part node from the shapes.multipart library you can write:

\documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{arrows.meta,
                calc,
                positioning,
                shapes.multipart}

\begin{document}
    \begin{tikzpicture}[
node distance = 3mm and 4mm,
base/.style = {minimum width=32mm, node font=\sffamily, align=center},
VMPN/.style = {% Vertical Multi Part Node
    rectangle split, rectangle split parts=5,
    draw},
 box/.style = {base, draw}
                        ]
\node (n1)  [VMPN]
            {\nodepart{one}     purchase (complain)
             \nodepart{two}     baggage (claim)
             \nodepart{three}   gates (unload)
             \nodepart{four}    runway (land)
             \nodepart{five}    airplane routing
             };
\node (n2)  [box, right=of n1.five east]    {airplane routing};
\node (n3)  [box, right=of n2]              {airplane routing};
\node (n4)  [VMPN, above right=0mm and 4mm of n3.south east]
            {\nodepart{one}     purchase (complain)
             \nodepart{two}     baggage (claim)
             \nodepart{three}   gates (unload)
             \nodepart{four}    runway (land)
             \nodepart{five}    airplane routing
            };
\node [above=of $(n2.north)!0.5!(n3.north)$]
    {\includegraphics[width=32mm]{example-image-duck}};
%
\node (n11) [base, below=of n1] {departure airport};
\node (n12) [base, below=of $(n2.south)!0.5!(n3.south)$]
                                {intermediate air-trafic\\control centers};
\node (n131) [base, below=of n4] {arrival airport};
%
\draw[blue!50!black, ultra thick, rounded corners=4mm, -{Triangle[angle=60:3pt 3]}]
    ([xshift=-3mm] n1.north west) |- ([yshift=-3mm] n12.south) -|
    ([xshift= 3mm] n4.north east);
    \end{tikzpicture}
\end{document}