Subscribe to a different ISP. Sometimes you can get the ISP to change your IP address as well, though they'll tend to ask for a reason and "I want to pwn n00bs" probably isn't sufficient.
There is nothing private about an IP address; your PC and other devices all share it with every website or server that you connect to; it's a requirement, because otherwise it would be like sending mail with no return address. A Game server can't send your game client information about the game state or changes in the game unless it knows where to send it, which is why it needs your IP Address.
There are VPN services you can use as well, however aside from it being unlikely to get assistance with that here, typically the use of VPNs is banned in the Terms of Service of online game services, which, I suppose would accomplish your goal of not being recognized though it would do so through a IP and possible even hardware fingerprint ban, depending on the particulars of the game.