There are a few [activity] night phrases that are commonly used to mean "a evening spent having fun doing [activity], usually with friends or family." The important implications are that the activity is fun and social.
Game night is an evening playing games, typically indoor games like board games, card games or charades. Usually it's assumed that one person is hosting several friends at their home (or that a family is playing together in their shared home). Sometimes the name of the game is used: poker night or D&D night for example. Under quarantine, game night has moved from being in-person to on-line in many cases, and that's what your referenced article is about.
Game night can also refer to watching an evening sporting event, either live or on television, again with the implication that it's a shared activity (either because you're watching in a group or because you're going to discuss the result with friends the next day). Fight night is similar and involves watching boxing or mixed martial arts. In Canada, the weekly hockey television broadcasts are called Hockey Night in Canada again referring to fun and trying to imply that watching hockey is a socially shared activity across the entire country.
Other constructions like this: movie night, date night, girls' night (meaning an evening of female friends having fun together). You can even use the name of a family meal (e.g., spaghetti night) with the implication that this meal is a particular favorite of most of the family.