I would do the whole thing with TikZ/PGF keys. If you want \Anton and \Berta available generally, you can use \tikzset{} in the preamble or at the beginning of the document.
There are various ways to do this. The way I've done it here involves setting up \Anton to store a simple value, defining an additional style (double trouble) to create \Berta from an argument using \pgfmathsetmacro, and then simply forwarding the value given to Anton to double trouble any time Anton is set.
This is more complicated to explain than to see in the code, I think!
\documentclass[tikz]{standalone}
\usepackage[utf8]{inputenc}
\tikzset{%
% key Anton stores a value in \Anton
Anton/.store in=\Anton,
% key double trouble sets \Berta to twice the argument it is given
double trouble/.code={%
\pgfmathsetmacro\Berta{int(2*#1)}%
},
% any time the key Anton is used, the same value will be passed to double trouble
Anton/.forward to=/tikz/double trouble,
% make sure the key Anton is set to something so \Anton and \Berta have some default value
Anton=15,
}
\begin{document}
\begin{tikzpicture}
[
x=1mm,
y=1mm,
mystyle/.style={<->},
every node/.style={fill=white},
]
\node (A) {A};
\node (B) at (\Anton,0){B};
\node (C) at (-\Berta,0){C};
% Maße Vertikal
% we can make this a bit more concise ...
\path [mystyle] (A) edge node {\Anton} (B) edge node {\Berta} (C);
\end{tikzpicture}
\end{document}

The advantage of using TikZ keys is that the user interface is intuitive and that it is easy to modify the values between or within tikzpictures.
Adding the following after the above picture, for example, we change the value of both macros by setting Anton=10 and then modify them again for just one node, setting Anton=2, so that \Berta produces 4 while \Anton still produces 10 when creating the nodes on the final line.
\begin{tikzpicture}
[
x=1mm,
y=1mm,
mystyle/.style={draw=blue, thick, text=red},
every node/.style={fill=white},
Anton=10,
]
\node (A) {A};
\node (B) at (\Anton,0){B};
\node (C) at (-\Berta,0){C};
% Maße Verti
% we can make this a bit more concise ...
\path [mystyle] (A) edge node {\Anton} (B) edge node [Anton=2] {\Berta} (C);
\end{tikzpicture}

Note that the distances used are all based on the initial value. The macros differ only when creating the node containing \Berta.
Whether this is useful and how useful depends, obviously, on the real use case. But it the kind of thing which is often required in TikZ pictures and it is nice to keep that flexibility, even if it is not of any immediate use.