I have no clue how the following gives you any matches...
specifically the presence of the "$" after the title= causes the entire pattern to always fail since that isn't the end of the string.
update pcilms_assign set intro=regexp_replace(intro, '/([^/]*)\" title=$', '&files=\1') where intro like '%https://owncloud.porterchester.edu%' and course=18 and id=55413;
and the result puts the &file= in the wrong place (at the end of the whole string).
The basic problem is that entirety of the content that your pattern matches is replaced with the totality of the replacement expression. Since you are matching the literal "title=" you have to somehow place that same literal in the result. You can capture it and then use "\2" or you can place it literally like Félix shows.
Alternatively, don't capture it. The way you match something without capturing it is by using what is termed a "zero-width" expression or a "look-around". In this case you want to "look-ahead" which is expressed thusly: (?=)
