A web browser does not simply just download and serve html anymore. Browsers have many features additional features that can be detected such as which version of flash your running, java and all add-ons you have installed which is can act as a very good fingerprint that be used to track people.
From the data collected 1 in 286,777 browers are unique, so from a legal point it would not stand in court.
You can test your brower uniqueness below
The electronic froniter foundation has an excellant paper on browser uniqueness