While an IP address may not be enough to identify you personally, there are strong correlations of various degrees, and in most cases, those correlations are only one step away. By itself, an IP address is just a number. But it’s trivial to find out who is responsible for that address, and thus who to ask if you want to know who it’s been given out to. In some cases, the logs will be kept indefinitely, or destroyed on a regular basis - it’s entirely up to each individual organization.Link
Up until now, I’ve only discussed the implications of having an IP address. The situation gets much much worse when you start using it. Because every bit of network traffic you use is marked with your IP address, it can be used to link all of those disparate transactions together.
Despite these possible correlations, not one of the major search engines considers your IP address to be personally identifiable information. While this may technically be true if you take an IP address by itself, it is a highly disingenuous position to take when logs exist that link IP addresses with computers, physical locations, and account information… and from there with people. Not always, but often. The inability to link your IP address with you depends always on the relative secrecy of these logs, what information is gathered before you get access to your IP address, and what other information you give out while using it.