The difference is that the average household shares multiple devices behind that single IP address, whereas IPv6 addresses are unique to the device you're using.

Identification to the level of IPv4 can still be done with IPv6 by using the /64 where you would previously take the /32, but with IPv6 you also get identifiers from within the network as well.

With how much IPv6 space is available, I'm not sure why SLAAC-based networks don't just assign different IP addresses to different use cases. I can see this becoming a problem on large company networks, but in home networks you could generate a random IPv6 address every hour for every website you visit and still never run out of address space.

Operating systems aren't exactly geared up for per-application outgoing IP addresses, and perhaps handling tens of thousands of IP addresses will bog down the kernel somehow, but in terms of privacy protection we could be doing a lot more than what IPv6 Privacy Extensions are doing right now.