If the browser created the unique identifier instead of the server it could create a new one as often as necessary.

I think that would actually work very nicely.

If you don't store a leakable cookie on the client and only use a connection ID (which is autoassigned by the server and cannot be easily forged) then a lot of cross-domain evil hackery goes away.

Of course, but the question is what that unique identifier denotes.

An IP address tells people who you are, where you are, and what you do.

A unique identifier ('+18El1iZRFCIiqRpfw4dJR8mXjJn2UxPrjwoRNpjSWg=' for instance) tells what you do without the who and where [1].

It's even possible advertisers could still target users based upon these identifiers, just without the background knowledge that makes these sorts of things privacy issues.

[1] I'll admit, it could be argued that the what can determine the who and where