I'm not an expert at this subject, but after some experimentation and detective work, I've found the following:
Your guesses about \UnicodeMathSymbol{"022B0}, \prurel and \mathrel seems to be based on the content of the file [...]texmf-dist/tex/latex/unicode-math/unicode-math-table.tex, which contains the following line:
\UnicodeMathSymbol{"022B0}{\prurel}{\mathrel}{element precedes under relation}%
That file is part of the package unicode-math.
According with the documentation of that package:
A simple beginning is:
\usepackage{unicode-math}
\setmathfont{xits-math.otf}
So I tried the following MWE:
\documentclass{article}
\usepackage{unicode-math}
\setmathfont{xits-math.otf}
\begin{document}
$a\prurel b$, $c ⊰ d$
\end{document}
which had to be compiled with xelatex or lualatex (because it uses .otf fonts, only supported with those engines), and produced:

The conclusion is then that the macro in question (\prurel) is simply a way to get the symbol from an appropiate math font which contains it, which in this case is xits-math.otf (but the documentation of unicode-math lists other compatible fonts), and it will not work if you don't have an appropiate font.
In addition, since apparently that symbol is implemented only in otf fonts, you would require xelatex or luatex to compile your document. I don't know if this qualifies as "collateral damages" :-)