That was not the point. It was a question about the rules and the set of rules that is applicable here is general privacy rules and the games EULA.
Actually, it's not that clear cut. It's not just the EULA. It's also the laws of whatever state or country each person is in you need to worry about. In your location it might be a 1 party consent environment. But that other person who told you not to record them? In their state or country it might be an all-party consent environment. In which case, if you didn't tell the other party you're recording you're breaking the law. Or if they ask you not to record them, and you do so anyway without telling them, this could be a problem. Especially if you verbally or in text agreed not to record.
A video game is not the same thing as being out in a public park. All parties have a reasonable expectation of privacy and not being recorded. In addition, your recording or streaming content may well run afoul of that game's EULA too. It may allow recording/streaming, but your specific content may be disallowed.
All of these reasons and more are why common decency says to ask before you record, and to not do so if anyone objects.
Lo, a quest! I seek the threads of my future in the seeds of the past.