There was a good answer about naming angles here: https://tex.stackexchange.com/a/55555/227339
I try to run it on my computer, but I get these errors:
As I understood, the problem is that name path=circle inside the scope is local, so I need to use name path global instead.
And yes, this code compiles and runs:
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}
\newcommand\markangle[6][red]{% [color] {X} {origin} {Y} {mark} {radius}
% filled circle: red by default
\begin{scope}
\path[clip] (#2) -- (#3) -- (#4);
\fill[color=#1,fill opacity=0.5,draw=#1,name path global=circle]
(#3) circle (#6mm);
\end{scope}
% middle calculation
\path[name path=line one] (#3) -- (#2);
\path[name path=line two] (#3) -- (#4);
\path[%
name intersections={of=line one and circle, by={inter one}},
name intersections={of=line two and circle, by={inter two}}
] (inter one) -- (inter two) coordinate[pos=.5] (middle);
% bissectrice definition
\path[%
name path=bissectrice
] (#3) -- (barycentric cs:#3=-1,middle=1.2);
% put mark
\path[
name intersections={of=bissectrice and circle, by={middleArc}}
] (#3) -- (middleArc) node[pos=1.3] {#5};
}
\begin{document}
\begin{tikzpicture}
\coordinate[label=below left:$A$] (A) at (0,0);
\coordinate[label=below right:$B$] (B) at (2,0);
\coordinate[label=above:$C$] (C) at (5,5);
\draw[thick] (B) -- (A) -- (C) -- cycle;
\markangle{A}{B}{C}{$\beta$}{5}
\markangle[blue]{B}{A}{C}{$\alpha$}{6}
\markangle[green]{B}{C}{A}{$\gamma$}{12}
\end{tikzpicture}
\end{document}
But if I'm right, name math global forces me to use unique path names through the entire document. That is really inconvenient inside large projects.
Then I remember: the original code from the answer doesn't have "global" option! It means, it can be compiled, but why do I get errors?

scope. That is reason why the second example work as is desired. – Zarko Dec 01 '20 at 15:44circlein a later picture again for another path. Of course, if you usecirclelater without redefining it you may get funny intersections. – Dec 01 '20 at 16:24(A)and then another coordinate also named(A), the first one will be "lost", too. If you want to have unique names, you can introduce counter, but I am not sure what the benefit in this context would be. – Dec 01 '20 at 17:02