They’re referencing the Fallout video game franchise. There’s a government referred to as the New California Republic, and their flag has a bear with two heads. Two-headed creatures mutated by radiation is kind of a running joke in the Fallout universe.
So. Fun fact. I've seen this tumblr post before (I mean given the amount of shit I have saved that's probably not a surprise) but I also had the exact same experience when I saw the post for the first time. I did not know that it was a real flag. I mean I'm Canadian so I have that excuse at least but damn.
I mean the multiheaded eagle has been used on many flags dating further back than the 1800s. So, wouldn't be too absurd for there to be a flag with another double headed animal on it I suppose. Just not sure there is a such a bear in any mythologies those founders may have subscribed to.
Here's an example in the wild that I remember. Anime reviewer Glass Reflection (13:39) talking about an anime/gaming convention in California that he attended. He's Canadian so he gets a pass. No hate, just a funny mistake.