String/vector concatenation symbol

Personally, I would use \oplus for this. I would think of it as a direct sum of two vectors (as a member of the direct sum of two vector spaces).

You can use two pluses joined together, which is rather intuitive

\documentclass[a4paper]{article}
\newcommand{\concat}{\ensuremath{+\!\!\!\!+\,}}
\begin{document}
$a \concat b$
\end{document}

Another symbol is ||

 \[ p\|q=p b^{l(q)}+q \]

The \| command gives a double vertical bar.

Since a vector in R^2 is written (1, 2), it would make sense to concatenate two vectors a and b as (a,b).