This is one of the things that I will have to personally take care of myself very soon, but I already got the basic gist from it from a lawyer, and essentially the whole thing is about what data you can associate with an individual person.
Things like email addresses, real names (not necessarily nicknames), credit card numbers, etc, are all things which bind data to a specific person, so those must be fully visible to their owner by law and should be comprehensible in the way they are worked with and in the way they are handled or shared with other entities.
IP addresses do not seem to be directly affected by this, not necessarily at least, over the simple fact that an IP address can be shared across an unlimited number of individuals, so it's a bit of a gray area for them.
At most, with the IP alone you can only identify the country of origin in many cases, or if you dig deep enough, the ISP, city, or maybe the set of clients who ever used that IP as generally you don't always have the same IP. Sometimes you cannot identify anyone at all.
However, it can also be said that a client with a fixed IP, or a client who was the the sole user of that IP for a specific amount of time as ISPs generally have this logged in some way, can be considered to be personal data that identifies an individual, so it's tricky, and I would say it depends on the situation, and how much data are mashed together in order to be able to identify an individual.
Also, from what I understood, the law can be enforced without having to delete all data from a user (when the user wants to be "forgotten"), because in some cases that might even be impossible.
However, deleting the actual data that identifies a user so that whichever other data there is no longer is connected to said user, is also a valid way of enforcing it, because now that data is associated with no one, and that's the whole concern about data protection here.
Therefore, an advised way to do it, is to never include personal data anywhere at all, and instead to actually have some sort of id that you generate yourself, and makes sense only to yourself, that you associate all that data with.
And then, from there, you would have a central place with all the actual IPs, emails and whatnot, which would be associated with that id.
Therefore should someone want to know all their data, you can query through that id you generated, and in case a user wants to have it forgotten, you just delete the email, IPs, etc themselves, while keeping the rest of the data intact if you wish, as the id no longer identifies "anyone".
It all goes down to whether or not you can say if data X belongs specifically to an individual or not. If the answer is no, there's no problem, if the answer is yes, you have to make arrangements so it can be both seen and "deleted" by that individual.
And in the case of game servers, I don't think that's an issue, because first and foremost you're not a business, and the businesses are the ones mostly targeted by this, and the IP resides in a gray area anyway, so it's not a focus of worry as long as the actual personal data is correctly taken care of, if existent at all, which in your case I don't think you have to worry about.