Since each user has a "profile", each browser is customized for each user.
Flash Player is a plug in so what you noted should happen.
I recently downloaded and installed Windows 10 onto my shared, Windows 7 computer. There are two accounts on that computer, mine and my wife's. After upgrading, my wife's "half" of the computer refuses to recognize that Adobe Flash Player is installed and demands that I download and install it. Yet, when I try, Adobe's website insists I already have it, and of course I do. My "half" of the computer works just fine -- no problems with Flash Player. What gives?