I found a random link on a forum linking to localbattle.net, at the same time I was running a local server on localhost. When I went to localbattle.net, my local server page appeared! The certificate was invalid, and the "Your connection is not private" warning screen appeared. Can that site see the content of my local server (page)? And is it a my client's issue, their server's issue, or my local server's issue?
This domain is used by Blizzard (battle.net) for some communication between their web pages and a server running in localhost.
Thus, when the web page wanted to communicate with the local install, it would direct a request to https://localbattle.net:22886/ were the game would be running a webserver. As they owned the domain, they would have issued a publicly valid certificate for that domain, which could then be embedded in the game, while they couldn't have obtained a certificate for
Do note however that this was still the wrong thing to do. Since the private key needed to be shared within the game, the private key isn't really private, and such certificate MUST be revoked by the CA within 24 hours of knowing about it (section 126.96.36.199 of the Baseline Requirements).
The public certificate running locally was originally reported by Tavis Ormandy and -after Hanno Böck reported it to mozilla and the CA- the certificate got revoked, after which Blizzard issued an update that installed a local (non-CA) certificate into the root trust store. Then they briefly used a new certificate, issued by digicert which also got revoked, and added to the OneCRL.
The whole story is explained quite well on this mozilla.dev.security.policy thread. If you want to dig up more, you can also read this Reddit thread, and the Blizzard post explaining their change to use a self-signed certificate.
$ dig 188.8.131.52 localbattle.net ... localbattle.net. 86400 IN A 127.0.0.1
The above shows that the name resolves globally to 127.0.0.1, i.e. localhost. It is unknown why the setup is this way and the question does not provide sufficient context to find out.