Why is there a green border around the annotation?

Use concept color=blue or draw=blue to override the inherited color settings from the mindmap nodes.

\documentclass[12pt]{article}
\usepackage[ngerman]{babel}
\usepackage[a4paper, text={16.5cm, 25.2cm}, centering]{geometry}
\usepackage[sfdefault]{ClearSans}
\usepackage[utf8]{inputenc}
\setlength{\parskip}{1.2ex}
\setlength{\parindent}{0em}

\usepackage{tikz}
\usetikzlibrary{mindmap}

\begin{document}


\begin{tikzpicture}[mindmap, grow cyclic,
  every node/.style=concept,concept color=lime!80, level 1/.append style={level distance=5cm, sibling angle=120}, level 2/.append style={level  distance=3 cm, sibling angle=60},
  every annotation/.style={concept color=blue,fill=blue!20, text width={}, align=left}%
]
  \node{Lehramt}
  child{node{Pädagogik}
    child{node{Steop}}
    child{node{Erziehen und Beraten}}
    child{node{Lehren und Lernen}}
    child{node{Schul\-architek\-tur}}
  }
  child{node{Sport}
    child{node{Physio\-logie}}
    child{node{Anatomie}}
    child{node{Inklusives}}
  }
  child[level distance=6cm]{node{Französisch}
    child{node{Sprach\-kurs}}
    child{node (n2) {Fach\-didaktik}}
    child{node{Sprach\-wissen\-schaft}} 
    child{node{Landes\-wissen\-schaft}}
    child{node{Medien\-wissen\-schaft}}
    child{node{Literatur\-wissen\-schaft}}
  };
  \node[annotation, right] at (n2.east) {Französisch unterrichten lernen};
\end{tikzpicture}
\end{document}