An
identifier which would uniquely identify a particular installation of Flash
Would be a privacy risk, aka ‘super-cookie’: like a browser cookie you couldn't refuse or delete.
Some controls (such as Windows Media Player) have had these before, and have had to be fixed. I know of no such thing in the Flash Player, but if there is a sneaky way to do it, it's not going to be a deliberate API you can rely on.
You can put an identifier in Flash Local Storage, which is in effect almost as good because most people don't realise that option exists, so few turn it off. (And if you do, many sites break. And even if you delete the stored values, Flash still locally remembers all the sites you've been to with Flash. It is not a very privacy-conscious plugin.)