Is there a reference managing software that shows/extracts content of highlights?
Citavi has the mentioned functionality. 

There are (at least?) four types of highlighting: Direct quotation (what you are describing - in blue), indirect quotation (pink), summary (green) and comment (not shown, same as in standard PDF viewers). All of them can navigate to their original location when clicking on the link-icon:



I downloaded one of the apparently most popular papers of 2017 and marked three random parts. The titles copy (part of) your marked text per default, but you can change it if you want.

I think it's free for up to 100 references. But if your school has some sort of partnership program, you might be able to get a full license. 

I switched from Mendeley to Citavi a few years ago. It was a little confusing at the beginning as I was expecting it to be "like Mendeley, but better", so I even used both parallel for some time. There are still a lot of functions I have no idea about, but documentation seems very thorough. At my school, there are even courses on how to use it.