not sure I understand your uniqueness condition, either
user account?
machine?
browser brand?
a combination of these above?

you could:
- have a hidden field in the page that you send with AJAX requests
- add a query string parameter to all URLs internal to the site (encoded as a GUID to avoid identity thief)

>Is there some kind of universal signature I could get from a browser, when the server processes the hit, so I would have an identifier that would allow me to do specific processes I need to do.
>
>For example, all Firefox installations on all PCs would have an identifier on each server request they send so if someone is using two instances of the same browser (on the same PC), it would then send the same identifier.
>
>We have this situation that a user is using two Firefox, using two logins, and because the cookie is send at the browser level, this creates some unexpected situations on the other instance when he switches back to it.
>
>As cookie as browser based, I cannot rely on that.
>
>As our application contains some specific options that should only be accessed by specific users, this also creates additional situations we have to negotiate with.