To me it is always helpful to color everything, so I can see what is happening. Your example is missing an intersection:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{angles,quotes,intersections}
\begin{document}
\begin{tikzpicture}
\coordinate (origin) at (0,0);
\draw [thick] (-2,2) coordinate (topy) -- (2,-2.5) coordinate (line1);
\draw[thick] (-3,-2) coordinate (line2) -- (3,2);
\pic [draw, -, "$\theta$", angle eccentricity=1.5] {angle = line2--origin--line1};
\end{tikzpicture}
% Improved example
\begin{tikzpicture}
\draw[thick,name path=A,red] (-2,2) coordinate (A) -- (2,-2.5) coordinate (B);
\draw[thick,name path=B,green] (-3,-2) coordinate (C) -- (3,2) coordinate (D);
\path[name intersections={of=A and B,name=inter}];
\draw pic["$\theta$", draw=blue, angle eccentricity=1.5] {angle=C--inter-1--B};
\end{tikzpicture}
\end{document}
See your attempt at the top and mine below:
