To generate the symbol shown in your second screenshot, you need to change the four \draw commands that are part of the tikzpicture environment. And, since you mention that it is supposed to represent a "twisted equal" symbol, I would assign it status mathrel.

Your tikz code assumes, at least implicitly, that the symbol will only ever be used in \displaystyle or \textstyle math mode, but not in \scriptstyle mode, let alone \scriptscriptstyle mode. Please advise if that's not your intention.
\documentclass[12pt]{report}
\RequirePackage{tikz}
\usepackage{amsmath} % for 'gather*' environment
\newcommand{\eq}{\mathrel{\begin{tikzpicture}%
[scale=.175, line width=0.5pt]
\draw (-1,-1) -- (0,0);
\draw (-1,0) -- (-0.5,-0.5);
\draw (0,-1) -- (1,0);
\draw (0.5,-0.5) -- (1,-1);
\end{tikzpicture}}}
\begin{document}
\begin{gather*}
u = v \\
u \eq v
\end{gather*}
\end{document}
Addendum: If the two parts of the symbol should be snugged up a bit, I'd use the following code:
\newcommand{\eq}{\mathrel{\begin{tikzpicture}%
[scale=.175, line width=0.5pt]
\draw (-1,-1) -- (0,0);
\draw (-1,0) -- (-0.5,-0.5);
\draw (-0.3,-1) -- (0.7,0);
\draw (0.2,-0.5) -- (0.7,-1);
\end{tikzpicture}}}

`, they'll be marked as code, as can be seen in my edit. You can also highlight the code and click the "code" button (with "{}" on it).+and-) or a relational operator (such as=and\ne)?