Welcome to the forum, ferral-cat!
I just checked that site and got the same message as you. In my testing of numerous other flash sites, flash player is actually working in Firefox and in Mint, but it seems that one particular site has an issue. So, your system is functioning properly, but that website is not. Until that site gets its act together, you may have to enjoy viewing others which are similar. Only rarely do I get those sites which will not work in Firefox/Mint. I just wanted to let you know the problem is not with your system.
As for the shockwave site, it said I did not have it either. I have never seen that message before on any site I have visited. My system is pretty standard, so not sure why it is not working, but then again, I have never seen that message. Not sure what to make of it, either.