This conversation is about Vampire: The Masquerade in particular, noting that it’s standard practice to name Sabbat packs but not Camarilla coteries, but it includes some thoughts on group naming in general.
And in general I don’t do it or see it unless a setting actively encourages it. The problem of coming up with a term that sums the group up means it hardly ever happens.
In twenty-two years of various World Of Darkness games I’ve been in two where it happened, Werewolf: The Apocalypse and Changeling: The Dreaming, where giving things significant names is part of the culture (and often done on a whim in the latter case).
Superhero groups need names because superhero groups in existing universes have them by default, but superhero game groups often come up short on inspiration. There are only so many good ones - hence the Avengers vs. the Avengers, and how many different Justice Leagues there are. We ended up being called the Super Squad in one Marvel game because we hadn’t come up with anything else after three months and we never did come up with anything better. And there was another Super Squad in the Marvel Universe (not to be confused with the Super Hero Squad) and of course it’s one of the names The Mystery Men rejected. In the game I’m currently playing I came up with a name for the villains we’re pretending to be (the Public Enemies, which I also think of as the title of the series but that might just be me) but got stuck on a hero team name.
The only Buffy group I’ve ever seen named, or even strongly nicknamed like the Scooby Gang, was a band. Because, again, bands need names. And a band name doesn’t have to have great significance when adventuring, like a starship name in a space opera game it doesn’t define the PCs as strongly.
The group in The Watch House was occasionally referred to as the Prentices (their proper title), the Watch or the Watchers, but no “team name” was ever proposed in six and a half years. This continues in other Actual Play games, using game-specific versions of “the party” but no real names, so the PCs in a Doctor Who game are “the travellers” and in Call of Cthulhu it’s “the investigators”. Specific named groups like the Theron Marks Society exist, but I’ve never seen the PCs set one up. Pendragon has a few orders of knighthood other than the Round Table, with the suggestion that the PCs might form one... but surely most PC knights would want to be on the Round Table too.
But, as noted in the thread, some players do this and expect it - a D&D game at Penny Arcade included a Wizards writer, who brought up the subject of a party name early on.
So, how about you?