> I tried this with emacs -Q on a mac using Gnu Emacs 26.3.
>
> Using customize-variable I set next-error-highlight to t ("Semipermanent
> highlighting") and saved that value for the current session.
>
> I then ran M-x grep and got a *grep* buffer and moved through the errors
> using n and p. The matches appeared in another window and were
> highlighted but the highlighting disappeared after a short time (could
> be the default value of 0.5 seconds) even if I didn't hit a key.
>
> My understanding of next-error-highlight is when set to t, the
> highlighting should remain until I enter another command. Yet 
> it seems to disappear on it's own.

Sorry for the delay, I looked at this bug report, and the solution for you
is to customize next-error-highlight-no-select instead of next-error-highlight,
because next-error-highlight-no-select affects the highlighting of
non-selected windows displayed from the *grep* buffer with n and p keys.

Before closing this bug report, I propose to improve the documentation: