That's embarrassing. I was hoping it was named after someone with the name of Bird and it was a sort of pun, but no. The original name for the house was the William Austin Bird House (after the architect William Austin). So when renaming it to Bird's World they did, in fact, just not know how basic punctuation works.
Yeah being named after a Mr/Mrs Bird would be quite a good name/pun. No such luck...
There's a mural just in the entrance that says "Welcome to A Bird's World" so if you were being generous maybe they're going for an exhibit where you experience what it's like to be a bird by entering a bird's world.
But then again on the map and on signs it's called "Bird's World" not "A Bird's World" so...