Reference different places to the same footnote

You could do this (tested in article and amsart):

Text with first footnote\footnote{\label{note1}This is the labeled footnote}
and more text with a second footnote\footnote{here}.

In this new paragraph we have a reference to the first
footnote\footnotemark[\ref{note1}].

The result will be

Edit: Further testing reveals, that the above solution fails when footnotes are numbered with anything other regular numbers. This is due to the fact, that the argument to \footnotemark is the value of a counter. Placing

\makeatletter
\newcommand\footnoteref[1]{\protected@xdef\@thefnmark{\ref{#1}}\@footnotemark}
\makeatother

in the preamble will allow us to mend the above example allowing any numbering of the footnotes.

Text with first footnote\footnote{\label{note1}This is the labeled footnote}
and more text with a second footnote\footnote{here}.

In this a paragraph we have a reference to the first footnote\footnoteref{note1}.

Edit 2: This will cause a problem when using hyperref where the link created by the \footnoterefmacro points to the previous footnote and not to the referenced footnote. Using the package cleveref as in this answer seems to be the perfect fit.

Objection, your honor:

Have you ever seen that in a book? Probably not. Because: Two footnotemarks numbered with e.g. an "8" raise the question whether it is a typo. But your readers can not come and ask you, if you really meant to do it this way.

Please, if you'd like to refer to the same text, make a footnote saying (maybe in better English than mine):

¹ Long sentence at: Wallace, Infinite Jest, p. 1234.

² See footnote 1.

What you'd like to do is against an old and sensible tradition. It neither an excuse that a part of Wikipedia does it, nor that it is technically possible.

I'm a bit surprised that noone mentioned KOMA-Script's \footref before. The KOMA-Script manual also has an example where multiple reference to the same footnote could make sense:

Maybe you have to mark each trade name with a footnote which states that it is a registered trade name.

One could think of other similar cases…

\documentclass{scrartcl}
\begin{document}

Company SplishSplash\footnote{This is a registered trade name.
All rights are reserved.\label{refnote}}
produces not only SplishPlump\footref{refnote}
but also SplishPlash\footref{refnote}.

\end{document}

This functionality can be used with the standard classes by using scrextend:

\documentclass{article}
\usepackage{scrextend}
\begin{document}

Company SplishSplash\footnote{This is a registered trade name.
All rights are reserved.\label{refnote}}
produces not only SplishPlump\footref{refnote}
but also SplishPlash\footref{refnote}.

\end{document}

The documentclass memoir already has its own \footref:

\documentclass{memoir}
\begin{document}

Company SplishSplash\footnote{This is a registered trade name.
All rights are reserved.\label{refnote}}
produces not only SplishPlump\footref{refnote}
but also SplishPlash\footref{refnote}.

\end{document}

And for completeness' sake: the footmisc package also defines \footref:

\documentclass{article}
\usepackage{footmisc}
\begin{document}

Company SplishSplash\footnote{This is a registered trade name.
All rights are reserved.\label{refnote}}
produces not only SplishPlump\footref{refnote}
but also SplishPlash\footref{refnote}.

\end{document}