I am trying to understand the relation between the demographics of Facebook users and the demographics of those using a specific hashtag in their posts.
As far as I know, you can not collect the Facebook data about demographics of users. However, you can use the Netvizz application for extracting the pages and groups data.
I have the link of this website that you can use, I don't know exactly if they fetch used hashtag's data from both facebook and twitter or not, but they are amazing when it comes to search for hashtags.
You can use an excel plugin called NodeXL (available here: https://nodexl.codeplex.com/releases/view/117659) it cost around $30 for students license but you can use it to collect data about hashtags on Twitter, Facebook, and Flickr, Youtube, (they are going to include Instagram and Redit too, but not sure when though).
If you want to collect data from Twitter only then you can also try GoogleTAGS, (available here: https://docs.google.com/spreadsheets/d/1EqFm184RiXsAA0TQkOyWQDsr4eZ0XRuSFryIDun_AA4/edit?pli=1#gid=8743918) (its free)
In addition to both of these tools there are many libraries in Python (e.g., tweepy, here are some documentations https://marcobonzanini.com/2015/03/02/mining-twitter-data-with-python-part-1/) that you can use to collect data from Twitter if you are comfortable with programming in python. I hope this helps!