diagonal text in math mode w/ vertical equality
I can't say I'm enamored of the look, but it does the first form you mention. The other can be done if you really want it, still. The key for the diagonal text is a recursive use of \brlap, which is a bottom-right lap, with the vertical offset of each lap scaled back, currently, to 38% of \baselineskip.
EDITED to allow horizontal kerning between letters to be set by \myhkern and vertical offset of successive letters to be set by \myvgap. In addition, \diagunder takes an optional argument (default =) defining the character rotated and placed below the first argument. Set as [] in order to begin the labels at the height where the default = sign would have been.
REEDITED to provide both \diagunder and \diagover, with identical syntax. Note that the diagonal text retains its height, but is perceived to have zero width, so that you will notice that (A,B) line up vertically in both over and under stacks.
\documentclass{article}
\usepackage{stackengine, graphicx}
\def\myhkern{.8pt}
\def\myvgap{.38\baselineskip}
\def\stacktype{L}
\newcommand\diagunder[3][\raisebox{-5pt}{$=$}]{%
\ifx\relax#1\relax%
\stackunder{$#2$}{\expandafter\angletext#3\relax}%
\else
\stackunder{$#2$}{\stackunder[.7\baselineskip]{\rotatebox{90}{$#1$}}{%
\expandafter\angletext#3\relax}}%
\fi
}
\def\angletext#1#2\relax{\small\setstackgap{L}{\myvgap}%
\kern\myhkern#1\angletextB#2\relax}
\def\angletextB#1#2\relax{%
\brlap{\kern\myhkern#1\ifx\relax#2\relax\else\angletextB#2\relax\fi}%
}
\newcommand\diagover[3][\raisebox{-5pt}{$=$}]{%
\ifx\relax#1\relax%
\stackon{$#2$}{\expandafter\upangletext#3\relax}%
\else
\stackon{$#2$}{\stackon[.9\baselineskip]{\rotatebox{90}{$#1$}}{%
\expandafter\upangletext#3\relax}}%
\fi
}
\def\upangletext#1#2\relax{\small\setstackgap{L}{\myvgap}%
\def\myslant{\kern\myhkern#1}\upangletextB#2\relax}
\makeatletter
\def\upangletextB#1#2\relax{%
\global\protected@edef\myslant{\tllap{\myslant}\kern\myhkern#1}%
\ifx\relax#2\relax\myslant\else\upangletextB#2\relax\fi%
}
\makeatother
\begin{document}
$(\diagunder{A}{alpha},\diagunder{B}{beta})$ versus
$(\diagunder[]{A}{alpha},\diagunder[]{B}{beta})$
$(\diagover{A}{alpha},\diagover{B}{beta})$ versus
$(\diagover[]{A}{alpha},\diagover[]{B}{beta})$
\end{document}
Adapting to the full MWE of the OP (some changes to my logic had to occur because my MWE above was stacked in text mode, while the OP's full example was stacked in math mode):
\documentclass{minimal}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tabstackengine}
\usepackage{mathtools}
\stackMath
\makeatletter
\renewcommand\TAB@delim[1]{\scriptstyle#1}
\def\req{\protect\rotatebox{90}{$\scriptstyle=$}}
\def\rne{\protect\rotatebox{90}{$\scriptstyle\ne$}}
\makeatother
\def\myhkern{.3pt}
\def\myvgap{.3\baselineskip}
\def\stacktype{L}
\def\mss#1{\mathrm{\scriptscriptstyle#1}}
\newcommand\diagunder[3][\scriptstyle=]{%
\ifx\relax#1\relax%
\stackunder{\scriptstyle#2}{\expandafter\angletext#3\relax}%
\else
\stackunder{\scriptstyle#2}{\stackunder[.7\baselineskip]{%
\rotatebox{90}{$#1$}}{\expandafter\angletext#3\relax}}%
\fi
}
\def\angletext#1#2\relax{\setstackgap{L}{\myvgap}%
\kern\myhkern\mss{#1}\angletextB#2\relax}
\def\angletextB#1#2\relax{%
\brlap{\kern\myhkern\mss{#1}\ifx\relax#2\relax\else\angletextB#2\relax\fi}%
}
\newcommand\diagover[3][\scriptstyle=]{%
\ifx\relax#1\relax%
\stackon{\scriptstyle#2}{\expandafter\upangletext#3\relax}%
\else
\stackon{\scriptstyle#2}{\stackon[.7\baselineskip]{\rotatebox{90}{$#1$}}{%
\expandafter\upangletext#3\relax}}%
\fi
}
\def\upangletext#1#2\relax{\setstackgap{L}{\myvgap}%
\def\myslant{\kern\myhkern\mss{#1}}\upangletextB#2\relax}
\makeatletter
\def\upangletextB#1#2\relax{%
\global\protected@edef\myslant{\tllap{\myslant}\kern\myhkern\mss{#1}}%
\ifx\relax#2\relax\myslant\else\upangletextB#2\relax\fi%
}
\makeatother
\begin{document}
\[
\setstackgap{L}{.6\baselineskip}\setstacktabulargap{0pt}\TABbinary
\tabularCenterstack{rcl}{
(&\diagover{A}{alpha}& , &\diagover{B}{beta}&) \hspace{1mm}= \hspace{1mm} F\\
&\rne&\\ &\mathclap{(\diagunder[]{D}{delta} , B)}&}
\]
\end{document}
And if you decided that you wanted to do it in \textstyle rather than \scriptstyle (since a lot of the math spacing, e.g. around = is more natural), then here is your MWE converted for that.
\documentclass{minimal}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tabstackengine}
\usepackage{mathtools}
\stackMath
\makeatletter
\def\req{\protect\rotatebox{90}{$=$}}
\def\rne{\protect\rotatebox{90}{$\ne$}}
\makeatother
\def\myhkern{.3pt}
\def\myvgap{.3\baselineskip}
\def\stacktype{L}
\def\mss#1{\mathrm{\scriptscriptstyle#1}}
\newcommand\diagunder[3][=]{%
\ifx\relax#1\relax%
\stackunder[.8\baselineskip]{#2}{\expandafter\angletext#3\relax}%
\else
\stackunder[.8\baselineskip]{#2}{\stackunder[.8\baselineskip]{%
\rotatebox{90}{$#1$}}{\expandafter\angletext#3\relax}}%
\fi
}
\def\angletext#1#2\relax{\setstackgap{L}{\myvgap}%
\kern\myhkern\mss{#1}\angletextB#2\relax}
\def\angletextB#1#2\relax{%
\brlap{\kern\myhkern\mss{#1}\ifx\relax#2\relax\else\angletextB#2\relax\fi}%
}
\newcommand\diagover[3][=]{%
\ifx\relax#1\relax%
\stackon[.8\baselineskip]{#2}{\expandafter\upangletext#3\relax}%
\else
\stackon[.8\baselineskip]{#2}{\stackon[.8\baselineskip]{\rotatebox{90}{$#1$}}{%
\expandafter\upangletext#3\relax}}%
\fi
}
\def\upangletext#1#2\relax{\setstackgap{L}{\myvgap}%
\def\myslant{\kern\myhkern\mss{#1}}\upangletextB#2\relax}
\makeatletter
\def\upangletextB#1#2\relax{%
\global\protected@edef\myslant{\tllap{\myslant}\kern\myhkern\mss{#1}}%
\ifx\relax#2\relax\myslant\else\upangletextB#2\relax\fi%
}
\makeatother
\begin{document}
\[
\setstackgap{L}{.85\baselineskip}\setstacktabulargap{0pt}\TABbinary
\tabularCenterstack{rcl}{
(&\diagover{A}{alpha}& , \diagover{B}{beta}) = F\\
&\rne&\\ &\mathclap{(\diagunder[]{D}{delta} , B)}&}
\]
\end{document}
Fun and reccursive :
$(\setstackgap{S}{-1pt}
\DiagUnder<$(\DiagUnder{D}{delta},B)$>{
\setstackgap{S}{3pt}%
\DiagUnder<\NEgal>{\DiagOn<\Egal>{A}{alfpha}}{}
}{},\DiagOn<\NEgal>{B}{beta})=F$
\documentclass{article}
\usepackage{tikz,xstring,ifthen,stackengine,xparse}
\usetikzlibrary{calc}
\def\R{String}
\def\N{0}
\newsavebox{\MyBox}
\makeatletter
\NewDocumentCommand{\DiagUnder}{
d<>
O{(-45:.8em)}
m
m
D<>{l}
}{%
\def\stackalignment{#5}%
\def\useanchorwidth{T}%
\def\@R{#4}%
\def\@N{0}%
\savebox\MyBox{
\begin{tikzpicture}[baseline=(x0),
every node/.style={inner sep=0pt,outer sep=0pt}]
\node (x0) at (0,-.16) {} ;
\whiledo{\NOT\equal{\@R}{}}{%
\StrLeft{\@R}{1}[\@L]
\StrGobbleLeft{\@R}{1}[\@R]
\node at ($(x0) + \@N*#2 $) {\strut\@L} ;
\pgfmathtruncatemacro\@N{\@N+1}
}
\end{tikzpicture}
}
\IfNoValueTF{#1}{% sans
\stackunder{$#3$}{\hspace{-3pt}\usebox{\MyBox}}%
}{% avec
\def\stackalignment{c}%
\stackunder{$#3$}{%
\def\stackalignment{l}%
\setstackgap{S}{1pt}%
\stackunder{%
\makebox[1em]{#1}}{\hspace{-2pt}\usebox{\MyBox}}}%
}%
}%
\NewDocumentCommand{\DiagOn}{
d<>
O{(-45:.8em)}
m
m
D<>{r}
}{%
\def\stackalignment{#5}%
\def\useanchorwidth{T}%
\setstackgap{S}{2pt}
\def\@R{#4}%
\def\@N{0}%
\savebox\MyBox{
\begin{tikzpicture}[baseline=(x0),
every node/.style={inner sep=0pt,outer sep=0pt}]
\node (x0) at (0,.16) {} ;
\whiledo{\NOT\equal{\@R}{}}{%
\StrLeft{\@R}{1}[\@L]
\StrGobbleLeft{\@R}{1}[\@R]
\node at ($(x0) + \@N*#2 $) {\strut\@L} ;
\pgfmathtruncatemacro\@N{\@N+1}
}
\end{tikzpicture}
}
\IfNoValueTF{#1}{% sans
\stackon{$#3$}{\hspace{-2pt}\usebox{\MyBox}}%
}{% avec
\def\stackalignment{c}%
\stackon{$#3$}{%
\def\stackalignment{r}%
\setstackgap{S}{-1pt}%
\stackon{%
\makebox[1em]{#1}}{\usebox{\MyBox}}}%
}%
}%
\makeatother
\newcommand{\Egal}{\,\rotatebox{90}{$=$}}
\newcommand{\NEgal}{\rotatebox{90}{$\neq$}}
\begin{document}
Bob$(\DiagUnder<\Egal>{A}{Diagonale},
\DiagUnder<\NEgal>[(-35:.9em)]{B}{Diagonale},
\DiagUnder[(-.2,.1)]{C}{Diagonale},
\DiagOn<\Egal>{D}{Diagonale},
\DiagOn<\NEgal>[(115:-.9em)]{E}{Diagonale},
\DiagOn[(.2,.1)]{F}{Diagonale}<l>)$
\end{document}