They can already do that, see e.g. what the UK is doing in response to tweets. You don't need identity verification to have an ISP tell you the person behind an IP.
IP addresses don't have anything like a 1:1 mapping to human beings and it's pretty trivial and inexpensive to get one from someone other than your ISP (e.g. use a VPN) if you have any concerns about that sort of thing.