The browser prompts the user to allow Flash if it detects the content requires Flash. If the content's Flash detection scripts don't conform to the browser requirements, the browser can't detect the content requires Flash and won't prompt the user to allow Flash. See Google's tech note, https://support.google.com/chrome/answer/6258784 for assistane.
Flash Player is a browser plugin, it will not display in the list of applications
Firefox and Chrome use different plugin technologies. Firefox usese NPAPI plugin, and Chrome uses PPAPI plugin. They are not interchangeable. Sounds like you may not have the NPAPI plugin installed. To install the NPAPI plugin, launch Firefox and navigate to https://get.adobe.com/flashplayer. The download page has OS/browser detection scripts to offer the appropriate plugin for the OS/browser is use.