In Austria it is since the introduction of data retention. Every time someone is assigned an IP address by his or her ISP, an entry is made so IP addresses can be mapped back to the person at any given time. It's pretty much the same across the EU, I reckon. I haven't heard of any cases yet where this data was used in a court of law though, but it is theoretically possible. Why would there be a law to oblige ISPs to do that if not for using this data in lawsuits? Back in 2006 when the EU guideline was made the official version was the usual terrorism bullshit (data is only usable for the prosecution of severe criminal action). In April this year the EU decided that file sharing is severe enough.
A side fact: Data retention hasn't proven to be very successful yet.
That would log an IP to a computer that is it. If I'm at your house and jump on your computer and download the latest and greatest movie. That would be logged as you doing that not me. The ISP would only have one piece of the puzzle, hence the problem with just tracking IP address.
A side fact: Data retention hasn't proven to be very successful yet.