I never even realised it was a thing people did commonly until I started watching Critical Role.
The closest my group has ever come is playing the Hell’s Rebels adventure path for Pathfinder. In that you resurrect an old group called the Silver Ravens, so that essentially becomes your party’s name.
I’ll preface this by saying that I haven’t played it in probably at least two decades, but I used to love Wacky Wheels. I believe it’s on Steam now, too.