A related idea to this:
A question prompted by the campaign for a new UK national bird. Does your setting have a national bird?
As I sit here, over my shoulder is a framed set of stamps, a souvenir from our childhood trip to the US, of every state’s official bird and flower. I can tell just from a glance that the Cardinal is very popular - not the state bird of Arizona, where the football team named after it is currently based, but the bird of Illinois where it started, among seven states. (The less striking Western Meadowlark appears to be the runner-up with six.)
The symbolic motif of a nation, state, corporation, or clan should tell you something about the group in question, even if it tells you it doesn’t matter to those involved as it was chosen by committee or has a meaning lost to history. This could be particularly true in a fantasy setting where animals and birds might have uncanny powers.
The Dire Wolf of House Stark is a real presence in Westeros, a symbol that became physical and important as the story began, Others are less significant - no stag appears to Robert Baratheon to tell him of his kingship, for example. The Flayed Man of House Bolton retains its historical significance (and really should have told the Starks something...) while the Dragon defines House Targaryen.
(See also Game Of Brands for modern corporations, and how some things never change.)