How do you find out what Firefox Profile is running when Firefox is running?

Enter "about:support" as URL and select the "Open containing folder" button. You can reach it using Menu "Help" -> "Troubleshooting Information", too:

(Screenshot by lifehacker.com)

Enter about:profiles in the URL address bar.

Using FF version 68.0.2, still no good built-in way to see the profile for the current browser session.
The best way I've found is to make an image with the text of the profile name and add about 150px of right padding. Then go into Add-ons, Themes, and at the bottom is a link to "Build your own theme with Firefox Color" (this takes you to https://color.firefox.com/). In there I add the Color addon and then select a preset theme and for Custom Background I select the image I created and select "No repeat". The text then shows in the top right next to the toolbar buttons. It's painful to setup because you need to do it for each profile you have but once it's done it works good.