With Windows 8 & 8.1, Flash Player is embedded by Microsoft... for Internet Explorer only. It must be updated through Windows Update. There are Windows files that prevent updating it from here. If however, you use another browser, then you can update it with the Flash Player Plug-in (All other browsers), but if you use Google Chrome, it has its own built in, like IE does now.
Windows 8.1 unfortunately, installs IE11, which is the worst browser in computing history because of three words:
That doesn't mean a lot, I'm sure, but it's the root of your problems, and Flash Player has nothing to do with it.
Microsoft "rewrote" the User-Agent Strings for the abomination they call their latest and greatest browser (Read more here).
User-Agent Strings are what websites use to identify the browser you're using and provide the proper content for it's browser engine, like ActiveX stuff, and Flash or HTML5 video.
Thanks to the geniuses in Redmond, WA, the User-Agent Strings for IE11 (which has a Trident engine), ID it as either "Gecko" (Firefox) or "Webkit" (Chrome).
Trouble is: when the site the directs to the content for one of these two engines, the Trident engine in IE can't intepret it and the site then sees IE as an "unidentified" browser.
The problem with an unidentified browser is that the plug-ins in that browser aren't recognized either, so even though you're up to date, it says you need the latest Flash Player when you use IE11.
YouTube... has converted to HTML5 video so if it doesn't detect Flash Player, it can display HTML5 (MP4) video which requires no plug-in to play.
Facebook can't do that, because HTML5 doesn't apply to games... only video.
Microsoft has no plans to "fix" the mess they've created because they think it's a great idea to block you out of content at the websites you visit and applications you use.
They recommend using "Compatibility View" and pretending that you're using an older verison of IE...
The Problem with THAT is that it's seen limited success, and you have to enable it for EVERY page that has problems... individually.
I'm not big on "pretending" so I recommend actually using another browser.
ANY of those will work where IE11 won't, with the Flash Player Plug-in (For all other browsers), and Chrome doesn't even need that, because it has its own Flash Player plugin built in.