How are secret identities less real than the extravagant fantasy elements?
I depends on the character. It brings a sense of humanity to them. Think of celebrities. We may know their real names, but do we really know how they are in their private lives?
While the superhero community and the government know that Banner is the Hulk, I doubt the average person knows or thinks about it. Like the old man who saw the Hulk fall from the sky.