It would also get its own unique IP, this jail.
Then I fire up firefox inside that chroot jail and use it to visit some websites ... and then I can wipe the whole thing out and redeploy again later, starting from scratch.
I don't need to trust incognito mode, I don't need to trust wiping cache or tabs talking to each other (or not) and I can worry a lot less about browser level exploits.
I can even put in firewall rules so that my "banking" instance can only talk to boa.com and scotttrade.com (or whatever).
It's totally workable (and I have done it) with vmware. Make a "bank browsing" VM and revert to pristine snapshot every day. The problem is that this is terribly heavyweight and overkill when I don't need a full blown VM.
It's not even really a browser issue - the real issue is, how do you jail a GUI application in X such that that window is in a chroot jail, distinct from the rest of your desktop?