For instance, I've heard lawyers said that a gorilla would not be permitted to testify in court, because we don't permit it. Or that secret identities would never be permitted.
The reason we don't permit animals to testify is that they don't talk in human-intelligible languages, and we don't have any translators. If we did. . .
All right, if there were a slow advent of powers, including Animal Speech, there would probably be an intermediate stage, where the superhero got on the stand and testified to what the animals said, just as a cadaver-dog handler would get to testify about the dog going wild at a location. But if they are clearly translating, it would probably end with animal witnesses.
As for secret identities, well, if someone tried to keep one nowadays, and was told he couldn't, he could say, "In that case, I'm leaving," and we could either let him, because we didn't care that much, or force him to reveal his identity.
Now try that on Superman.
The first point is that you want him on your side much more than you want any run of the mill character. Supervillains come around, you know you can't stop them by yourself. And just as important, if Superman leaves, you can't stop him. You are probably going to have to give him some concessions to keep him on your side.
Vigilantes would probably, of course, be those most insistent on it, but if they have the powers and are useful, they can obviously get concessions. Which will doubtlessly annoy those who want the law to treat people equally, but the inequailty is inherent in the superpowers. . . (This is a special case of what Jordan S. Bassior discusses in Metahumans as Ultimate Weapons )