If you don't want to use hyperref (i) make sure your sections are
numbered (e.g. #+OPTIONS: num:5) and (ii) make sure that you are not
giving links a description.  Here's an example:

* test my heading
:PROPERTIES:
:CUSTOM_ID: t
:END:
[[*test my heading][my funky label]]
[[*test my heading]]
[[#t]]

→

\section{test my heading}
\label{sec-1}
\hyperref[sec-1]{my funky label}
\ref{sec-1}
\ref{sec-1}

—Rasmus

--
May the Force be with you