Your browser gives you away without any cookie. Whether yours is unique is not something you have to take on faith: you can look.
A page draws text and symbols into an invisible image and turns the pixels into a number. The same with a tone nobody hears. Add fonts, screen size and core count. Together that yields a value that recognises a browser for days — without anything being stored.
Open one of the public fingerprinting tests, for example Cover Your Tracks by the EFF. Look at the result without an extension, then install AdSilence and reload. The difference is right in front of you.
With AdSilence, Cover Your Tracks reports a browser whose fingerprint changes — instead of a value you can be found by. This is built against mass trackers, not against someone hunting for you specifically.