My project does that.[1] Since each profile has its own profile.js and extensions. One application is that I hate triggering financial portals to re-authenticate. So all of my finance happens from my home IP, no matter where I am. And instead of a heavy VPN, it is just an ssh tunnel. When I was in university, my school browser profile always routed through my school proxy. If I start using facebook again, I will use Facebook's TOR public gateway, but only for the Facebook profile.
[1]: https://github.com/unqueued/foxbox