No, it's not propagated to other browsers signed into the same account, it explicitly says that in the settings page.
Maybe part of the issue is that I am logged into the same account across three different devices, or that I have other Google accounts too logged in (but I don't use YouTube in them, only the account with Premium).
Anyway, it should be trivial for Google to make it actually permanent (I would love it it was stored in my account, which I pay for Premium on, instead of local or cookie storage), but lots of people have this problem.
