In C there is no confusion between 
struct Point {
    int x;
    int y;
};
and
union Point {
    int x;
    int y;
};
which are two different types called struct Point and union Point respectively.
The C99 standard section 6.7.2.1 states:
6  Structure and union specifiers have
  the same form. The keywords struct and
  union  indicate that the type being
  specified is, respectively, a
  structure type or a union type.
7 The
  presence of a struct-declaration-list
  in a struct-or-union-specifier
  declares a new type,   within a
  translation unit.
So it most unequivocally declares a type. The syntax for type names in C is given in sections 6.7.6 and includes the specifier-qualifier-list from 6.7.2, which takes the form of struct-or-union identifier.   
Does this code works in C99? Or is this a "C++ thing"?
No, C99 does not decide to promote structure types over enum types and union types with the same name. It is a "C++ thing", as struct and classes are mostly the same thing in C++, and classes are important to C++.