Firebase won't register user information

Solution 1:

I see two mistakes.

First, you should use the modular syntax for your auth state handler:

useEffect(() => {
  onAuthStateChanged((user) => { // 👈 Remove the auth. here
    setUser(user);
  });
});

And second, the default value of the user is an object, not an array:

const [user, setUser] = useState({});

There may be more problems, but these jumped out at me.