In my Amherth setting, players are expected to name their parties because it helps them gain recognition and future job offers. City criers are paid to shout out the deeds of famous adventuring companies in public forums (markets, taverns, courts, etc.) and these companies tend to become popular with the public.
I've noticed that choosing a name, colors, symbol, etc. actually helps the players become more focused and can often set the tone of the game (for good or ill).
Some players come up with a theme before play (like the Greatswords* where everyone carries two-handed weapons), some choose a name they think sounds cool (like the Society of the Holy Brand*), while others pick up a name after their first few (mis)adventures (like the Duck Slayers*).
Pete
*yes, these were the names of actual adventuring companies. . .