Arrow bending in TikZ is not aligned

One thing that does usually not get appreciated enough is that curved paths with arrows get distorted unless you are loading the bending library. This is true regardless of whether or not you actually add the bend key to the arrow heads, see p. 204 of pgfmanual v3.1.5

Yet I would also bend the arrow heads. And I would choose another way to bend the arrows: with the in and out syntax.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{arrows.meta,bending,positioning} 
\tikzset{
    %Define standard arrow tip
    >={Stealth[bend]},
    %Define style for boxes
    punkt/.style={
           rectangle,
           rounded corners,
           draw=black, very thick,
           text width=6.5em,
           minimum height=2em,
           text centered},
    plain/.style={
           rectangle,
           rounded corners,
           draw=white, very thick,
           text width=6.5em,
           minimum height=2em,
           text centered},
    % Define arrow style
    pil/.style={
           ->,
           thick,
           shorten <=2pt,
           shorten >=2pt,}
}

\begin{document}
    \begin{tikzpicture}
     \node[punkt] (a) {AAAAA};

     \node[punkt, inner sep=5pt, right=0.5 of a] (b) {BBBBBB BBBBB}
       edge[pil,<-] (a.east);

     \node[plain, inner sep=5pt, right=0.5 of b.east] (dummy) {};
     \node[punkt, inner sep=5pt, above=0.5 of dummy] (b1) {BBBBBB BBBBB1}
       edge[pil, <-, out=180,in=60] (b.north);
     \node[punkt, inner sep=5pt, below=0.5 of dummy] (b2) {BBBBBB BBBBB2}
       edge[pil, <-, out=180,in=-60] (b.south);

     \node[plain, inner sep=5pt, right=0.5 of dummy] (dummy2) {};
     \node[punkt, inner sep=5pt, above=0.5 of dummy2] (c1) {CCCCCC CCCC1}
       edge[pil,<-] (b1.east);
     \node[punkt, inner sep=5pt, below=0.5 of dummy2] (c2) {CCCCCC CCCC2}
       edge[pil,<-] (b2.east);

     \node[punkt, inner sep=5pt, right=0.5 of dummy2] (d) {DDDDDD DDDD}
       edge[pil, <- , out=120,in=0] (c1.east)
       edge[pil, <- , out=-120,in=0] (c2.east);

    \end{tikzpicture}
\end{document}

You may liked: instead edge[bend ...] use \draw (x) |- (y) or \draw (x) -| (y) or \draw (x) -| (y) where paths have rounded corners:

\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{arrows,
                positioning}

\begin{document}
    \begin{tikzpicture}[
    node distance = 3mm and 5mm,
     punkt/.style = {draw, very thick, rounded corners,
                     text width=6.5em, minimum height=2em,
                     inner sep=5pt, align=center},
every path/.style = {draw, rounded corners=6mm, stealth-,
                     thick, shorten <=2pt, shorten >=2pt},
                        ]
\node[punkt] (a) {AAAAA};
\node[punkt, right=of a] (b) {BBBBBB BBBBB} edge (a);

\node[punkt, above right=of b] (c1) {BBBBBB BBBBB1};
       \draw    (c1) -| (b);
\node[punkt, below right=of b] (c2) {BBBBBB BBBBB2};
       \draw    (c2) -| (b);

\node[punkt, right=of c1] (d1) {CCCCCC CCCC1} edge  (c1);
\node[punkt, right=of c2] (d2) {CCCCCC CCCC2} edge  (c2);

\node[punkt, below right=of d1] (e) {CCCCCC CCCC1};
      \draw    (e) |- (d1);
      \draw    (e) |- (d2);
    \end{tikzpicture}
\end{document}